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Given any triangle ABC. Join vertices A,B,C with midpoints (diagonal intersections) say P, Q, R of squares constructed on opposite sides BC, AC and AB respectively. The lines AP, BQ and CR concur at a point say H.

It can be proved that this point is the orthocenter of triangle PQR.

Does the point H have any name or number in the original triangle ABC?

Arne Erikson
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  • You should include an illustration/diagram – 冥王 Hades Nov 25 '22 at 10:18
  • Perhaps someone could help? I have had problems uploading images to math stackexchange. – Arne Erikson Nov 25 '22 at 16:54
  • Please link the image here in the comments section. I'll edit it into the question – 冥王 Hades Nov 25 '22 at 17:00
  • It's Vecten Point (X(485) in ETC). – Intelligenti pauca Nov 25 '22 at 17:32
  • I am afraid I can't even make a link to the image here in the comments section. My image is a rather poor photo of a hand drawn construction, so it would be infinitely better if someone used to GeoGebra or some other geometric program tool could supply an image. By the way H is also the circumcenter of a different triangle made by constructing parallellograms on each of the neighbouring squares and joining the three points "opposite" the vertices. An article by D. Detemple and S. Harolds: " A round-up of square problems" has the details. – Arne Erikson Nov 25 '22 at 18:30
  • Shouldn't the comment above @intelligenti pauca be called an answer? It perfectly answers my question. – Arne Erikson Nov 25 '22 at 19:52

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