Posted on Saturday, June 6, 2015
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Unusual Properties of Acute Triangles
http://www.egyptorigins.org/
I list here some of the unusual properties of acute triangles. This will permit us to gain insight into why the ancient Egyptians displayed half-angles in their constructions. (Acute triangles are those in which the angles of the triangle are all less than ninety degrees. If an angle is larger than ninety degrees the triangle becomes obtuse.)
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If we take the angle bisectors in any acute triangle as shown above, these are the half-angles. We can see that if these are projected across the triangle from the three vertices they all are concurrent, that is, they meet at a common point within the triangle.
Similarly, if we project the side bisectors of the triangle from the respective vertices we see that they also are concurrent.
Third, if we project the perpendiculars of the triangle from the respective vertices we see that they too are concurrent.
But this is not the end of the unusual properties of acute triangles. Many others exist.
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Take the side bisector and the angle bisector shown in the drawing above, denoted by the red and blue lines respectively. Then create another line with an angle between it and the angle bisector the same as the angle between the side and angle bisectors. This is shown by the green line. Do this for the three vertices. One will find that the three new lines will also intersect at a common point. These lines are called the symmedians, and the point at which they are concurrent is called the Lemoine Point.
For more on Emile Michel Hyacinthe Lemoine (Nov 22,1840 - Dec 21,1912) see
One can find the Lemoine point by a different construction. If one draws a line from the mid-point of a side of the triangle to the mid-point of the perpendicular to that side, and thus for all three sides, the three new lines will be concurrent at the Lemoine point.
Lemoine's work in mathematics was mainly on geometry. He founded a new study of the properties of a triangle. In a paper of 1873 he studied the point of intersection of the symmedians of a triangle. He had been a founder member of the Association Française pour l'Avancement des Sciences and it was at a meeting of the Association in 1873 in Lyon that he presented his work on the symmedians.
A symmedian of a triangle from vertex A is obtained by reflecting the median from A in the bisector of the angle A. He proved that the symmedians are concurrent, the point where they meet now being called the Lemoine point. Among other results on symmedians in Lemoine's 1873 paper he showed that the symmedian from the vertex A cuts the side BCof the triangle in the ratio of the squares of the sides AC and AB. He also proved that if parallels are drawn through the Lemoine point parallel to the three sides of the triangle then the six points lie on a circle, now called the Lemoine circle. Its centre is at the mid-point of the line joining the Lemoine point to the circumcentre of the triangle.
A symmedian of a triangle from vertex A is obtained by reflecting the median from A in the bisector of the angle A. He proved that the symmedians are concurrent, the point where they meet now being called the Lemoine point. Among other results on symmedians in Lemoine's 1873 paper he showed that the symmedian from the vertex A cuts the side BCof the triangle in the ratio of the squares of the sides AC and AB. He also proved that if parallels are drawn through the Lemoine point parallel to the three sides of the triangle then the six points lie on a circle, now called the Lemoine circle. Its centre is at the mid-point of the line joining the Lemoine point to the circumcentre of the triangle.
For example, I measured the lengths of the sides in my original of the Symmedian drawing above. These were 7.81 inches for the left side, 5.09 inches for the right side, 4.91 inches from the left vertex to the crossing of the symmedian line with the base, and 2.09 inches from the crossing of the symmedian line on the base to the right vertex. The first numbers square to 60.996 and 25.908 respectively. The ratio of these two numbers is 2.35. The ratio of 4.91 to 2.09 is also 2.35, demonstrating the finding of Lemoine.
This is a diagram of the Lemoine circle. The green lines are the symmedians. The blue lines are the lines parallel to the respective sides, drawn through the Lemoine point. One can see how the Lemoine circle intersects where the parallel blue lines cross the three sides of the triangle. Since the blue lines are parallel to a side, they must cross the alternate triangle sides. That a perfect circle can be drawn through the six points seems uncanny.
Here I show the circumcircle of the triangle, with the Lemoine Circle Center. One can see that the Lemoine Circle Center is half the distance from the Lemoine Point to the Center of the Circumcircle. I do not offer a geometric proof of this fact, but from my measurements with QuickCad software I know this fact to be true.
Nearly a hundred years ago when Lemoine made his discoveries he had to do so through construction with ruler and compass. Computer power was not available to him. Many other interesting (to mathematicians) relationships exist within acute triangles. These relationships are now explored with computers. Clark Kimberling of Evansville University in Indiana has listed hundreds of interesting facts about acute triangles. See
Consider the Incircle. If we take the point where the angle bisectors of an acute triangle meet (are concurrent) as the center point of a new circle and draw that circle so that it just touches one side of the triangle we find that it will just touch all three sides of the triangle. This new circle is called the Incircle. (Note that we are reverting to the angle bisectors to do this construction.) The perpendicular distances from the concurrent point to each side are equal.
Proof of various properties of the Incircle have been given by Ira Fine and Tom Osler of Rowan University at Glassboro, New Jersey. See their Paper. Please note that their paper is in PDF format. You will need Adobe PDF reader to view it. I post that paper with their permission.
If one examines the drawing of the Incircle, one can see how new equal triangles are formed at each vertex, with a common base determined by the (half) angle bisector. For example, the distance from the vertex A to point a is the same as the distance from the vertex A to point c. Since the distance from the center of the Incircle to the respective perpendicular base sides are equal, two similar triangles are formed, mirror images of one another. That is, A to a = A to c, a to Incircle Center = c to Incircle Center, and A to Incircle Center is the same for both triangles. Then by flipping A-c-Incircle Center over on the A-Incircle Center axis, one would obtain two congruent triangles. Similarly for B to a = B to b, and C to c = C to b.
The side bisectors, side perpendiculars, and symmedians do not provide similar relationships; the angle bisectors are unique.
Another unique point is determined for acute triangles. This is obtained by drawing lines from the respective vertices to where the Incircle perpendiculars touch the respective sides. The three lines are concurrent at the Gergonne Point. See:
See also discussion on Triangle Centers:
For short discussion on Gergonne see:
To be continued.
Why Half Angles?
Half angles are a fundamental property of circular geometry. They have been studied since time immemorial, and discussed by many geometricians.
The easiest way to describe a half-angle is to show it in the following diagram.
The semicircle DEF contains three triangles of interest.
First, is the right triangle. Regardless of where B may fall on DEF, it will always produce a right triangle, DBF.
Second is the right angle produced by ABC. A line CB is erected perpendicular to DF that just touches the circle at B. The hypotenuse AB defines this triangle. This triangle may or may not be a Pythagorean Triangle.
Third is the right angle produced by DBC. The same perpendicular line, CB, is used as one side of this third triangle. The hypotenuse DB defines this triangle. This triangle may or may not be a Pythagorean Triangle.
The angle, BDC, is always one-half the angle, BAC, regardless of where B may fall. I now proceed to demonstrate this fact.
Thales' Theorem
In geometry, Thales' theorem (named after Thales of Miletus) states that if D, B and F are points on a circle where the line DF is a diameter of the circle, then the angle DBF is aright angle.
Proof
We use the following facts: (a) the sum of the angles in a triangle is equal to 180°, (b) the base angles of an isosceles triangle are equal, and (c) the diameter of a circle expressed in angular measure is 180°.
Let A be the center of the circle. Since AD = AB = AF, ABF and ABD are the two isosceles triangles. (Each has two sides equal, and the two equal sides of each are equal to each other. However, their angles differ.)
(Equation 1):
These conditions make the angles of the isosceles triangle ABF = b + 2a = 180°, where b is the angle at the center (in this case) and a is one of the two equal angles.
(Equation 2):
Likewise for the isosceles triangle ADB, d + 2g = 180°.
Note that d = (180° - b). This takes advantage of the fact that the diameter of a circle in angular measure is 180°.
Then using the alternate expression for d, (180° - b), and substituting into Equation 2, we obtain
(180° - b) + 2g = 180°.
Simplifying we find that b = 2g.
Substituting back into Equation 1 we see that 2g + 2a = 180°.
Dividing by 2: g + a = 90°.
Q.E.D.
Corollary 1
Equation 2 above shows that b = 2g. This means that g is the half-angle of b. This is true regardless of where B may lie on the circle.
Corollary 2
From Equation 1 we know that 2a = (180° - b). But (180° - b) is the alternate angle to b. Hence, a = (180° - b)/2 = d/2. This means that a is the half-angle of d. This is true regardless of where B may lie on the circle.
Converse
I offer the following without proof.
The converse of Thales' theorem is also true. It states that if you have a right triangle and construct a circle with the triangle's hypotenuse as diameter, then the third vertex of the triangle will lie on the circle.
The theorem and its converse can be expressed as follows:
- The center of the circumcircle of a triangle lies on one of the triangle's sides if and only if the triangle is a right triangle.
The circumcircle is that circle that totally encloses the triangle. For a right triangle the diameter is the hypotenuse.
History
Thales was not the first to discover this theorem since the Egyptians and Babylonians must have known of this relationship. However surviving evidence does not provide evidence that they could prove the theorem, and the theorem is named after Thales because he was said to have been the first to prove the theorem, using his own results that the base angles of an isosceles triangle are equal, and that the sum of angles in a triangle is equal to 180°
As I have shown, the knowledge of the ancients was far more than empirical. Through the remnants of the mathematical documents available to us we know their level of knowledge was far greater than indicated in the Greek evidence.
Greek Proof
The Greek mathematician, Euclid of Alexandria, (c 325 - c 265 BC), in his Proposition 20, Book 3, gave a general proof that the half-angle statement is true. (Euclid labeled his points differently.)
The numbers 1.5 and 1.32 refer to other Euclid Propositions which serve to prove the missing part, and which you may find atEuclid Prop attached.
More Unusual Properties of Acute Triangles
The Nine-Point Circle of Triangle ABC with orthocenter H passes through:
1. the midpoints L, M, and N of the three sides,
2. the feet of the altitudes D, E, and F to those sides, and
3. the points X, Y, and Z, which are the midpoints of the segments AH, BH, and CH, respectively.
1. The Nine-Point Center U lies on the Euler Line of Triangle ABC.
The Euler line is the line passing through
2. the orthocenter H,
3. the circumcenter CC, and
4. the centroid G of a triangle.
The tangents to the Nine-Point Circle at the midpoints L, M, and N of the sides of the triangle form a triangle, RST, that is similar to the orthic triangle (the triangle DEF). In fact, the sides of this triangle are parallel to those of triangle DEF.
For an extensive listing of 20 additional properties about the nine point circle, see the following reference, specifically pages 53-56:
MacKay, J. S. (1892). History of the Nine Point Circle. Proceedings of the Edinburgh Mathematical Society, (11). pages 19-61.
Feuerbach's Theorem
The Nine-Point Circle of a triangle "touches" the incircle and the three excircles.
Altitude
In geometry, an altitude of a triangle is a straight line beginning at a vertex and ending perpendicular to (i.e. forming a right angle with) the opposite side, or an extension of the opposite side. The intersection between the (extended) side and the altitude line is called the foot of the altitude. This side is called the base of the altitude. The length of the altitude is the distance between the base and the vertex.
| Corollaries |
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| In an isosceles triangle (a triangle with two equal sides), the altitude having as base the third side will have the midpoint of that side as foot. |
| In a right triangle, the altitude with the hypotenuse as base divides the hypotenuse into two lengthsp and q. If we denote the length of the altitude by h, we then have the relation h2 = pq. |
The three altitudes of a triangle intersect in a single point, called the orthocenter. The orthocenter lies inside the triangle (and consequently the feet of the altitudes all fall on the triangle) if and only if the triangle is not obtuse (i.e. does not have an angle bigger than a right one).
Four points in the plane such that one of them is the orthocenter of the triangle formed by the other three are called an orthocentric system.
Centroid
In geometry, the centroid or barycenter of an object X in n-dimensional space is the intersection of all hyperplanes that divide Xinto two parts of equal moment about the hyperplane. Informally, it is the "average" of all points of X.
In physics, the centroid can, under some circunstances, coincide with an object's center of mass and also with it's center of gravity. In some cases this leads to the usage of those terms interchangingly. For a centroid to coincide with the center of mass, the object should have uniform density or the matter's distribution through the object should have certain properties, such assymmetry. For a centroid to coincide with the center of gravity, the centroid must coincide with the object's center of mass and the object must be under the influence of a uniform gravitational field.
Note that a figure's centroid need not necessarily lie within it; the centroid of a crescent, for example, lies somewhere in the central void.
The lines extending from each vertex of a triangle to the mid-point of the opposite sides are called medians.
The three medians of a triangle intersect at a common point. This is called the centroid of the triangle.
This point is also the triangle's center of mass, if the triangle is made from a uniform sheet of material
Symmedian
In geometry, three special lines are associated with every triangle, the triangle's symmedians. One starts with a median of the triangle (a line connecting one vertex with the midpoint of the opposite side) and reflects it at the corresponding angle bisector (the line through the same vertex that divides the angle of the triangle there in two equal parts). The resulting line is a symmedian. The three symmedians intersect in a single point, the triangle's symmedian point or Lemoine point.
Note that all three sets of lines, the median, the angle bisectors, and the symmedian are respectively concurrent.
A triangle with medians (blue), angle bisectors (green) and symmedians (red). The symmedians intersect in the Lemoine point L.
The symmedian point of a triangle with sides a, b and c has homogeneous trilinear coordinates [a : b : c].
The symmedian point L can also be constructed differently: the three lines joining the midpoint of a side to the midpoint of the altitude on that side intersect in L. The symmedian point of a right triangle is therefore the midpoint of the altitude on the hypotenuse.
The Gergonne point of a triangle is the same as the symmedian point of the triangle's contact triangle.
Circumcircles of Triangles
The circumcircle of a triangle is the unique circle on which all of its three vertices lie. The circumcenter of a triangle can be found as the intersection of the three perpendicular bisectors. (A perpendicular bisector is a line that forms a right angle with one of the triangle's sides and is erected from that side at its midpoint.) This is because the circumcenter is equidistant from any pair of the triangle's points. All points on the perpendicular bisectors are equidistant from the respective points of the triangle.
A triangle is acute (all angles smaller than a right angle) if the circumcenter lies inside the triangle. The triangle is obtuse (has an angle bigger than a right one) if the circumcenter lies outside the triangle. The triangle is a right triangle if the circumcenter lies on the hypotenuse. This is one form of Thales' theorem.
The diameter of the circumcircle can be computed as the length of any side of the triangle, divided by the sine of the opposite angle. (As a consequence of the law of sines, it doesn't matter which side is taken: the result will be the same.) The triangle's nine point circle has half the diameter of the circumcircle.
The circumcenter always lies on one line with the triangle's centroid and orthocenter. This line is known as Euler's line.
