![Rendered by QuickLaTeX.com ABC](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-b5002f05fd90c80f81a9e0e9c845e02d_l3.png)
![Rendered by QuickLaTeX.com \angle C=90^{\circ}](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-795fc88636925de3fb7f8773d7d30b4a_l3.png)
![Rendered by QuickLaTeX.com F_c](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-f7973a989a256020197983ef145840aa_l3.png)
![Rendered by QuickLaTeX.com C](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-3238eb6669792857eccb924606e71d82_l3.png)
![Rendered by QuickLaTeX.com K](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-45553b59c9922b4b50444b81c7a16595_l3.png)
![Rendered by QuickLaTeX.com S](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-8392836d00fdce623aa8536af20050bc_l3.png)
![Rendered by QuickLaTeX.com SK,KF_c, CK](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-ff789f0bb72fc79d325631afd17928d6_l3.png)
Place the vertices of the right triangle at convenient points: ,
, and
. Then:
is the point
;
is the point
;
is the point
.
![Rendered by QuickLaTeX.com C](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-3238eb6669792857eccb924606e71d82_l3.png)
Using the given coordinates, we find:
Using the given coordinates, we find:
![Rendered by QuickLaTeX.com C](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-3238eb6669792857eccb924606e71d82_l3.png)
Using the given coordinates, we find:
![Rendered by QuickLaTeX.com KF_c^2=(CK)(SK)](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-3a305e2ced0e8f1f37d8af50df532cf7_l3.png)
Follows from
![Rendered by QuickLaTeX.com CK=2SK](https://blog.fridaymath.com/wp-content/ql-cache/quicklatex.com-a8b00411b4f14ea791eaf83290c0d429_l3.png)
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