# Problem in an Equilateral Triangle

The following problem has been offered at the 1980 All-Union Russian Olympiad and published in Crux Mathematicorum a decade later (1990, 33 and 70):

A line parallel to the side AC of equilateral ΔABC intersects BC at M and AB at P, thus making ΔBMP equilateral as well. D is the center of ΔBMP and E is the midpoint of CP. Determine the angles of ΔADE.

### This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

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I came across this problem in one of R. Honsberger's books, where it was accompanied by an interesting commentary:

While the complete specification of D inside ΔBMP is well known, with E situated at the midpoint of CP, ΔADE is cast obliquely in ΔABC, and with the medians not meeting the sides of a triangle at distinguished angles, it is doubtful that comparing the directions of the sides of ΔADE with those of ΔABC will uncover much useful information. Of course, there is always the chance that there exists an elusive construction line which will make everything clear. Finding Euclidean geometry so attractive, I am always reluctant to throw in the towel on the synthetic approach; however, after several looks at the problem from this point of view, I turned to the analytic approach.

The applet is supposed to suggest what that "elusive construction line" might be that which would make everything clear. A different solution appears elsewhere.

### References

1. R. Honsberger, In Pólya's Footsteps, MAA, 1999, pp. 125-126 ### This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

 What if applet does not run?

The key observation here is that, as P travels over AB, the points E and D each trace a straight line. E stays on the midline E'E'' of ΔABC parallel to the base AB while D is always located on the angle bisector of angle B. With this realization at hand, the problem appears as a particular case of a more general problem of three similar triangles.

In two extreme positions ABE' and AOE'', where O is the center of ΔABC, we obviously have two ° triangles with 30°, 60°, 90° angles. If we could show that points D = D(P) and E = E(P) divide BO and, respectively, E'E'' in the same proportion, the theorem of three similar triangles would apply directly: the triangle ADE would be a linear combination of triangles ABE' and AOE'' and thus be similar to either. It would then have the same 30° - 60° - 90° angles.

But the required condition is easily obtained from a comparison of two configurations of similar triangles. E'E/EE'' = BP/PA from the similarity of triangles ABC and E"E'C, that of PBC and EE'C, and also the similarity of triangles APC and E"EC. On the other hand, BD/DO = BP/PA from the similarity of PBD and ABO. • Equilateral Triangles on Sides of a Quadrilateral
• Euler Line Cuts Off Equilateral Triangle
• Four Incircles in Equilateral Triangle
• Problem in Equilateral Triangle
• Problem in Equilateral Triangle II
• Sum of Squares in Equilateral Triangle
• Triangle Classification
• Isoperimetric Property of Equilateral Triangles
• Maximum Area Property of Equilateral Triangles
• Angle Trisectors on Circumcircle
• Equilateral Triangles On Sides of a Parallelogram
• Pompeiu's Theorem
• Circle of Apollonius in Equilateral Triangle
• 