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:
- the right triangle is isosceles
- the Kosnita point coincides with the centroid.
Task
- (Foot of the symmedian) Let
be a triangle having vertices at
,
,
. VERIFY that:
- the foot of the symmedian from
is
- the foot of the symmedian from
is equidistant from the feet of the altitudes from
and
.
- the foot of the symmedian from