Thread: geometry

prove that the circumcenter of a given triangle is the orthocenter of the triangle formed by joining the midpoints of the sides of the original triangle. thanks for ur help

Originally Posted by peri123

prove that the circumcenter of a given triangle is the orthocenter of the triangle formed by joining the midpoints of the sides of the original triangle. thanks for ur help

The original triangle is ABC.

The mid-parallels to the sides of the triangles form the new triangle.

The mid-parallel is parallel to (drawn in red). The perpendicular bisector of a is perpendicular to too. The midpoint of a is the vertex of the new triangle.

The perpendicular bisectors of ABC cut themselves in P. Since the perp. bisectors pass through the vertices of the new triangle and are perpendicular to the sides of the new triangle they are simultaneously heights in the new triangle. Since the bisectors of the original triangle intersect at P the heights of the new triangle intersect in P too.