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Unusual Properties of Acute Triangles

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.)

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.

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

http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Lemoine.html

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 Franšaise 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 BC of 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

http://faculty.evansville.edu/ck6/

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:

http://mathworld.wolfram.com/GergonnePoint.html

See also discussion on Triangle Centers:

http://mathworld.wolfram.com/TriangleCenterFunction.html

For short discussion on Gergonne see:

http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Gergonne.html

To be continued.

 

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