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# 9.6: Tangent half-lines

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Suppose $$ABC$$ is an arc of a circle $$\Gamma$$. A half-line $$[AX)$$ is called tangent to the arc $$ABC$$ at $$A$$ if the line $$(AX)$$ is tangent to $$\Gamma$$, and the points $$X$$ and $$B$$ lie on the same side of the line $$(AC)$$. If the arc is formed by the line segment $$[AC]$$ then the half-line $$[AC)$$ is considered to be tangent at $$A$$. If the arc is formed by a union of two half lines $$[AX)$$ and $$[BY)$$ in $$(AC)$$, then the half-line $$[AX)$$ is considered to be tangent to the arc at $$A$$.

Proposition $$\PageIndex{1}$$

The half-line $$[AX)$$ is tangent to the arc $$ABC$$ if and only if

$$\measuredangle ABC + \measuredangle CAX \equiv \pi$$.

Proof

For a degenerate arc $$ABC$$, the statement is evident. Further we assume the arc $$ABC$$ is nondegenerate. Applying Theorem 9.1.1 and Theorem 9.2.1, we get that

$$2 \cdot \measuredangle ABC + 2 \cdot \measuredangle CAX \equiv 0$$.

Therefore, either

$$\measuredangle ABC + \measuredangle CAX \equiv \pi$$,     or     $$\measuredangle ABC + \measuredangle CAX \equiv 0$$.

Since $$[AX)$$ is the tangent half-line to the arc $$ABC, X$$ and $$B$$ lie on the same side of $$(AC)$$. By Corollary 3.4.1 and Theorem 3.3.1, the angles $$CAX$$, $$CAB$$, and $$ABC$$ have the same sign. In particular, $$\measuredangle ABC + \measuredangle CAX \not\equiv 0$$; that is, we are left with the case

$$\measuredangle ABC +\measuredangle CAX \equiv \pi$$.

Exercise $$\PageIndex{1}$$

Show that there is a unique arc with endpoints at the given points $$A$$ and $$C$$, that is tangent to the given half line $$[AX)$$ at $$A$$.

Hint

If $$C \in (AX)$$, then the arc is the line segment $$[AC]$$ or the union of two half-lines in $$(AX)$$ with vertices at $$A$$ and $$C$$.

Assume $$C \not\in (AX)$$. Let $$\ell$$ be the perpendicular line dropped from $$A$$ to $$(AX)$$ and $$m$$ be the perpendicular bisector of $$[AC]$$.

Note that $$\ell \nparallel m$$; set $$O = \ell \cap m$$. Note that the circle with center $$O$$ passing thru $$A$$ is also passing thru $$C$$ and tangent to $$(AX)$$.

Note that one the two arcs with endpoints $$A$$ and $$C$$ is tangent to $$[AX)$$.

The uniqueness follow from Proposition $$\PageIndex{1}$$.

Exercise $$\PageIndex{2}$$

Let $$[AX)$$ be the tangent half-line to an arc $$ABC$$. Assume $$Y$$ is a point on the arc $$ABC$$ that is distinct from $$A$$. Show that $$\measuredangle XAY \to 0$$ as $$AY \to 0$$.

Hint

Use Proposition $$\PageIndex{1}$$ and Theorem 7.4.1 to show that $$\measuredangle XAY = \measuredangle ACY$$. By Axiom IIIc, $$\measuredangle ACY \to 0$$ as $$AY \to 0$$; hence the result.

Exercise $$\PageIndex{3}$$

Given two circle arcs $$AB_1C$$ and $$AB_2C$$, let $$[AX_1)$$ and $$[AX_2)$$ at $$A$$, and $$[CY_1)$$ and $$[CY_2)$$ be the half-lines tangent to the arcs $$AB_1C$$ and $$AB_2C$$ at $$C$$. Show that

$$\measuredangle X_1AX_2 \equiv -\measuredangle Y_1CY_2.$$ Hint

Apply Proposition $$\PageIndex{1}$$ twice.

(Alternatively, apply Corollary 5.4.1 for the reflection across the perpendicular bisector of $$[AC]$$.)