5 Tangents -- Insert a Tangent Line
I take the following approach (from Dorrie's 100 Great Problems)
Let X1, X2, X3, X4, X5 be the intersections of tangents (V, I), (I, II), (II, III), (III, IV), (IV, V).
Now suppse X1... X5 define a hexagram, Mr. Penty, where two vertices coincide (pass into a tangent line).
Working backward through Brianchon's and Pascal's theorems, we may
- Choose any point on the given tangents, and introduce it as a new vertex of a second hexagram inscribed about the same conic section, sharing four vertices with Mr. Penty
- 2. Draw the sixth vertex of the new hexagram, giving us a new tangent line.
- 3. Find the point of tangency on this new line.