For any point P inside a given triangle ABC, denote by x, y, and z the distances from P to the lines [BC], [AC], and [AB], respectively. Find the position of P for which the sum x^2 + y^2 + z^2 is a minimum.?
1 Answer
It's the Lemoine Point (Symmedian Point)
Explanation:
To construct this, draw the symmedians (the reflection of the medians by the bisections of the respective angles) and mark the intersection point.
Demonstration:
If AP is a symmedian then the distances to the sides AB and AC are proportional to the sides themselves.
So... The symmedian point would have the following property:
Knowing that we can start the proof.
We want some point that minimize the expression
So...
Which is the minimum value for the sum of three squares.
This would make