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 $(AC)$.

$截屏2021-02-19 上午9.44.12.png$

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.

$截屏2021-02-19 上午9.46.58.png$

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

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

$截屏2021-02-19 上午9.56.08.png$

Hint

Apply Proposition $\PageIndex{1}$ twice.

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

Proposition \(\PageIndex{1}\)

Exercise \(\PageIndex{1}\)

Exercise \(\PageIndex{2}\)

Exercise \(\PageIndex{3}\)