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Six Incircles in an Equilateral Triangle: What is it about?
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The applet purports to illustrate one of the many properties of an equilateral triangle.
Let M be a point inside an equilateral triangle ABC, with pedal points A', B', and C', as shown. The lines joining M to the six points split the triangle into six smaller ones. Circles are inscribed into each of the latter.
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If the diameter of the incircle of triangle UVW is denoted D(UVW), the applet suggests that
| (1) | D(AMB') + D(BMC') + D(CMA') = D(A'MB) + D(B'MC) + D(C'MA). |
For a proof, note that all six triangles are right, with the right angles at the pedal points A', B', C'. For a right triangle with legs a and b and the hypotenuse c, the inradius r is determined by
| (2) | 2r = a + b - c, |
see the diagram in the Proof #33 of the Pythagorean proposition. With (2) in mind, (1) reduces to
| (3) | AB' + BC' + CA' = A'B + B'C + C'A. |
Draw AbAc||BC, BaBc||AC, and CaCb||AB. This gets us three equilateral triangles with the pedal points A', B', C' as the midpoints of their bases. That is to say,
| (4) |
A'Bc = A'Cb B'Ca = B'Ac C'Ab = C'Ba. |
Substituting (4) into (3) and noting that, e.g.,
| (5) | ACa + BAb + CBc = BCb + CAc + ABa. |
Finally observe that
| (6) |
ACa = BCb BAb = CAc CBc = ABa, |
which proves (5).
References
- T. Andreescu, B. Enescu, Mathematical Olympiad Treasures, Birkhäuser, 2004
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