Euler line proof

Let O the circumcenterMathworldPlanetmath of ABC and G its centroid. Extend OG until a point P such that OG/GP=1/2. We’ll prove that P is the orthocenterMathworldPlanetmath H.

Draw the median AA where A is the midpointMathworldPlanetmathPlanetmathPlanetmath of BC. Triangles OGA and PGA are similarMathworldPlanetmath, since GP=2GO, AG=2AG and OGA=PGA. Then OAG=PGA and OAAP. But OABC so APBC, that is, AP is a height of the triangle.

Repeating the same argument for the other medians proves that P lies on the three heights and therefore it must be the orthocenter H.

The ratio is OG/GH=1/2 since we constructed it that way.

Title Euler line proof
Canonical name EulerLineProof
Date of creation 2013-03-22 11:44:29
Last modified on 2013-03-22 11:44:29
Owner drini (3)
Last modified by drini (3)
Numerical id 15
Author drini (3)
Entry type Proof
Classification msc 51M99
Classification msc 55U10
Classification msc 18E30
Classification msc 18-00
Classification msc 55U35
Classification msc 46-01
Classification msc 47B25
Classification msc 81-01
Related topic EulerLine