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Kosnita point in a right triangle II

Let be a right triangle in which , is the foot of the altitude from , is the Kosnita point, and is the symmedian point. Then the lengths of the segments form a geometric progression:

Place the vertices of the right triangle at convenient points: , , and . Then:

• is the point ;
• is the point ;
• is the point .
Find the distance from the Kosnita point to the foot of the altitude from .

Using the given coordinates, we find:

Find the distance from the Kosnita point to the symmedian point.

Using the given coordinates, we find:

Find the distance from vertex to the Kosnita point.

Using the given coordinates, we find:

PROVE that .

Follows from

PROVE that .

Follows from

Takeaway

In any right triangle, the following statements are equivalent:

1. the right triangle is isosceles
2. the Kosnita point coincides with the centroid.