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单词 EulerLineProof
释义

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.

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更新时间:2025/5/4 14:43:25