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A sample problem from social media I

Examine the following problem from a Facebook group:
Solve the above problem. (See below for enlarged image.)

Sample solution here .

Takeaway

Let be a non-right triangle with side-lengths , and let be as defined in today’s post. Then the following statements are equivalent:

1. .

• Let be a triangle with vertices at , , and . Let be as defined in today’s post.
1. Find the coordinates of
2. Verify that , where is the circumradius.

An example of harmonic division II

Let be a right triangle in which , and let be the foot of the altitude from . The segment contains two special points: the symmedian point (), and the Kosnita point (). We show an example where and divide harmonically.
Given with vertices located at , , and , find the symmedian point.

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 .

Given with vertices located at , , and , verify that and divide harmonically, where is the Kosnita point, is the symmedian point, and is the foot of the altitude from .

We have , , , and .

Explicit calculation gives

Thus and divide harmonically, since:

Takeaway

In any right triangle, the following statements are equivalent:

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