# Tangent to Circles - Problem Solving

One tangent line, and only one, can be drawn to any point on the circumference of a circle, and this tangent is perpendicular to the radius through the point of contact. In the figure below, line $BC$ is tangent to the circle at point $A$.

If two tangents are drawn to a circle from an external point,

- the lengths of the tangents from the external point to the points of contact are equal;
- they subtend equal angles at the center of the circle.

In the figure below, $AB=CB, \angle AOB= \angle COB$ and $\angle ABO=\angle CBO$.

In the figure below, $BA$ and $BC$ are tangents to the circle at points $A$ and $C$, respectively. If $\angle AEC=110^\circ$ and $\angle DCE=80^\circ$, find $\angle ABC$.

Consider the diagram to the right.

- Opposite angles of a cyclic quadrilateral sum up to $180^\circ$. So $\angle ADC=180^\circ-110^\circ=70^\circ.$
- From the inscribed angle theorem, we can see that $\angle AOC=2 \angle ADC=2(70^\circ)=140^\circ.$
Since the sum of interior angle of a quadrilateral is $360^\circ$, we have $\begin{aligned} \angle ABC &=360^\circ-\angle BCO-\angle BAO-\angle AOC\\ &=360^\circ-90^\circ-90^\circ-140^\circ\\ &=40^\circ.\ _\square \end{aligned}$

On the same side of a straight line three circles are drawn as follows:

- A circle with a radius of 4 cm is tangent to the line.
- The other two circles are identical, and each is tangent to the line and to the other two circles.

What are the possible radii of these two identical circles?

A.$\ \,$ 24 cm only

B.$\ \,$ 20 cm only

C.$\ \,$16 cm only

D.$\ \,$ 24 and 20 cm only

E.$\ \,$ 20 and 16 cm only

F.$\ \,$ 16 and 24 cm only

G.$\ \,$ 16, 20, 14 cm only

H.$\ \$ No solution

$$

**Note:** A diagram has not been provided because drawing the correct diagram is part of the problem solving.

$\triangle ABC$ at points $D, E, F.$

In the figure to the right, the yellow circle is tangent to$2 \sqrt{14}, 4 \sqrt{3},$ and $\sqrt{42}$ are the lengths of the common tangents to the semicircles of diameters $(BD, CD), (CE, AE),$ and $(AF, BF),$ respectively.

What is the radius of the yellow circle?

**Cite as:**Tangent to Circles - Problem Solving.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/tangent-to-circles-problem-solving/