Our main goal is to classify △ ABC by its angles and sides. As a part of the solution we need to draw △ ABC, as well as a diagram that illustrates the given situation. Let's make it. We will start by plotting a circle with a center at point P.
We are also given that segments AB, BC, and AC are tangent to ⊙ P at points X, Y, and Z, respectively. Therefore, we first need to add tangent lines to ⊙ P at points X, Y, and Z. Let's recall that a tangent line is perpendicular to a radius of the circle, so we need to draw radii XP, ZP, and YP.
Now we are ready to draw tangent lines to ⊙ P.
The points of intersection of the tangent lines correspond to points A, B, and C. Let's mark them on the diagram.
Let's color the segments of △ ABC in orange.
Recall that all angles in equiangular triangles are congruent and have a measure of 60 ^(∘). Therefore, all angles in △ XYZ measure 60^(∘).
We want to find the measures of the arcs intercepted by these angles. To find the measures of the arcs, we will use the Measure of an Inscribed Angle Theorem, Theorem 10.10.
m XZ = 2 m ∠ Y ⇒ m XZ &= 120 ^(∘)
&⇓
m XY &= 120^(∘)
m YZ &= 120^(∘)
By the definition of a central angle its measure is the measure of a minor arc. We obtain the following relations.
m ∠ XPZ &= 120 ^(∘)
m ∠ ZPY &= 120 ^(∘)
m ∠ XPY &= 120 ^(∘)
