How can we draw a triangle give one of its vertex and the orthocentre and circumcentre?
I tried to invoke the concept of 9 point circle and tried using the centroid but could not succeed in making it... Please help...
2026-04-02 16:26:27.1775147187
Coordinate-geometry curiosity question
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First locate the centroid and then draw the cevian through it from the given vertex, which should meet the opposite side at its midpoint, which can be found because the centroid divides the cevian in a ratio of 2 to 1. The opposite side is perpendicular to the altitude from the given vertex through the orthocentre, and the two vertices on that side can be found using the circumcircle.
Note also that the construction fails if the orthocentre is the same as the given vertex. Moreover, in that case there are indeed multiple solutions; in particular there is a solution for each choice of the second vertex on the circumcircle.