Axioms that aren't axiomatic
Which of the following statements is incorrect?
The orthocenter of a triangle is the incenter of it's orthic triangle.
The centroid of a triangle is also the centroid of the triangle formed by joining the centroids of the 3 smaller triangles obtained from the original one.
For an equilateral triangle inscribed in a circle, the sum of the 2 shorter distances from any point on the circle to the vertices of the triangle is equal to the distance from that point to the remaining vertex.
The side length of an equilateral triangle inscribed in a circle with radius equals .
Note - The orthic triangle is a type of pedal triangle which is obtained by joining the 3 points where the altitudes meet the larger triangle's sides.
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1 solution
Proof of ( A ) is as follows -
For a proof of ( B ) , see my solution to this problem .
( C ) is a special case of Ptolemy's theorem (It can be easily seen by joining each of the vertices to the point in question).
I found a proof of ( D ) here .