Elementary Geometry

Miscellaneous Propositions

Triangles Circumscribing a Triangle

Triangles of constant species circumscribed to a triangle. 977 Let 𝐴𝐵𝐶 be any triangle, and 𝑂 any point; and let circles circumscribe 𝐴𝑂𝐵, 𝐵𝑂𝐶, 𝐶𝑂𝐴. The circumferences will be the loci of the vertices of a triangle of constant form whose sides pass through the points 𝐴, 𝐵, 𝐶.

Proof

Draw any line 𝑏𝐴𝑐 from circle to circle, and produce 𝑏𝐶, 𝑐𝐵 to meet in 𝑎. The angles 𝐴𝑂𝐵, 𝐶𝑂𝐴 are supplements of the angles 𝑐 and 𝑏 (III.22); therefore 𝐵𝑂𝐶 is the supplement of 𝑎 (I.32); therefore 𝑎 lies on the circle 𝑂𝐵𝐶. Also, the angles at 𝑂 being constant, the angles 𝑎, 𝑏, 𝑐 are constant. 978 The triangle 𝑎𝑏𝑐 is a maximum when its sides are perpendicular to 𝑂𝐴, 𝑂𝐵, 𝑂𝐶.

Proof

The triangle is greatest when its sides are greatest. But the sides vary as 𝑂𝑎, 𝑂𝑏, 𝑂𝑐, which are greatest when they are diameters of the circles; therefore ⋯ by (III.31). 979 To construct a triangle of given species and of given limited magnitude which shall have its sides passing through three given points 𝐴, 𝐵, 𝐶.
Determine 𝑂 by describing circles on the sides of 𝐴𝐵𝐶 to contain angles equal to the supplements of the angles of the specified triangle. Construct the figure 𝑎𝑏𝑐𝑂 independently from the known sides of 𝑎𝑏𝑐, and the now known angles 𝑂𝑏𝐶=𝑂𝐴𝐶, 𝑂𝑎𝐶=𝑂𝐵𝐶, ⋯. Thus the lengths 𝑂𝑎, 𝑂𝑏, 𝑂𝑐 are found, and therefore the points 𝑎, 𝑏, 𝑐, on the circles, can be determined.
The demonstrations of the following propositions will now be obvious.

Sources and References

https://archive.org/details/synopsis-of-elementary-results-in-pure-and-applied-mathematics-pdfdrive