We have studied four triples of lines associated with a given triangle DeltaABC having sides of lengths a, b, c and vertex angles alpha, beta, gamma. The following table, which is provided for reference, lists homogeneous coordinates relative to the vertices of the triangle for the intersection point P of each of four triples of coincident lines.
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Previously the point was made that homogeneous coordinates of the point where the altitudes of a triangle with acute angles intersect relative to the vertices of the triangle are (proportional to) the areas of the three subtriangles formed by that point as third vertex with any side of the given triangle. This is a special case of the more general:
Theorem. Let A, B, and C be three non-collinear points, and let P = uA + vB + wC (u + v + w = 1) be a point inside DeltaABC, i.e., with u, v, w > 0. Then u, v, w are, respectively, the ratios of the areas of DeltaBCP, DeltaCAP, DeltaABP, respectively, to the area of DeltaABC.
Proof. By symmetry, it is enough to check that the area of DeltaABP is equal to w times the area of DeltaABC. Since AB is a common side in these two triangles, it is enough to check that the altitude length from P to AB is w times the altitude length from C to AB. Let F be the point where the line CP meets AB, let S be the foot of the altitude from P to AB, and let R be the foot of the altitude from C to AB. Then DeltaPSF is clearly similar to DeltaCRF. Therefore, the altitude length ratio |PS| / |CR| is equal to the hypotenuse ratio |PF| / |CF| . By the principle of preservation of proportionality in barycentric coordinates (u + v)F = uA + vB. Hence, P = (u + v)F + wC, and by the fulcrum principle |PF| / |CF| = w.
Let c, h, and q be given with c > 0 and h > 0. Let A, B, and C be the points in R^{2} defined by
| A = (0, 0) , B = (c, 0) , and C = (q, h) . |
Show that A, B, C are not collinear.
Find the three points where the altitudes of DeltaABC meets the sides of the triangle.
Find the point H where the three altitudes of DeltaABC meet.
Find the barycentric coordinates of H relative to the vertices A, B, C.
Find the tangents of the vertex angles in DeltaABC.
Do you see how to construe your calculations in the previous exercise as giving a proof that the angle tangents are homogeneous coordinates of the altitude intersection point in any triangle? In other words, is there anything special about the triangle of the previous exercise apart from its location?