Observe that the given triangle is right-angled at . Further, in a right triangle, the symmedian point is the midpoint of the right-angled vertex and the foot of the altitude from the right-angled vertex. In a previous post, we found the foot of the altitude from to be . Thus, the symmedian point in this case is .
We have , , , and .
Explicit calculation gives
Thus and divide harmonically, since:
- (Equivalent statements) Let be a non-right triangle. PROVE that the three statements below are equivalent:
- (Equal steps) Suppose that triangle satisfies any of the three equivalent statements above. In such a triangle, let be the foot of the symmedian from , and let be the feet of the altitudes from in that order. PROVE that .