Pillow-Problems: Problem #4

Problem #3 | Problem #5
For more information on this collection, see Pillow-Problems by Charles L. Dodgson (Lewis Carroll).

Problem:

In a given acute-angled Triangle inscribe a Triangle, whose sides make, at each of the vertices, equal angles with the sides of the given Triangle.

Solution:

Let ABC be the given Triangle, and A′B′C′ the required Triangle, so that ∠ BA′C′ = ∠CA′B′, &c.

Evidently, A′C′, A′B′ are equally inclined to a line drawn, from A′, ⊥; BC; and so of the others: i.e. these ⊥;s bisect the ∠s at A′, B′, C′;

∴ they meet in the same Point. Draw them; let them meet at O; and call the ∠C′A′B′ ‘2α’, and so on.

Now (β + γ) = π − ∠B′OC′ = A;

∴ 2A = 2(β + γ) = π − 2α

α = 90° − A;

∴ ∠BA′C′ = A.

Similarly ∠BC′A′ = C.

∴ Triangle BC′A′ is similar to Triangle BCA; and so on of the others;

∴ A′ is foot of ⊥ drawn, from A, to BC. Hence the construction is obvious.

Q.E.F.

Problem #3 | Problem #5
For more information on this collection, see Pillow-Problems by Charles L. Dodgson (Lewis Carroll).