9.5: Arcs of circlines

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A subset of a circle bounded by two points is called a circle arc.

More precisely, suppose $A, B, C$ are distinct points on a circle $\Gamma$. The circle arc $ABC$ is the subset that includes the points $A, C$ as well as all the points on $\Gamma$ that lie with $B$ on the same side of $(AC)$.

For the circle arc $ABC$, the points $A$ and $C$ are called endpoints. There are precisely two circle arcs of $\Gamma$ with the given endpoints; they are opposite to each other.

Suppose $X$ be another point on $\Gamma$. By Corollary 9.3.2 we have that $2 \cdot \measuredangle AXC \equiv 2 \cdot \measuredangle ABC$; that is,

$\measuredangle AXC \equiv \measuredangle ABC$ or $\measuredangle AXC \equiv \measuredangle ABC + \pi.$

$截屏2021-02-19 上午9.13.18.png$

Recall that $X$ and $B$ lie on the same side from $(AC)$ if and only if $\angle AXC$ and $\angle ABC$ have the same sign (see Exercise 3.4.2). It follows that

$X$ lies on the arc $ABC$ if and only if

$\measuredangle AXC \equiv \measuredangle ABC$;

$X$ lies on the arc opposite to $ABC$ if

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

Note that a circle arc $ABC$ is defined if $\triangle ABC$ is not degenerate. If $\triangle ABC$ is degenerate, then arc $ABC$ is defined as a subset of line bounded by $A$ and $C$ that contain $B$.

$截屏2021-02-19 上午9.15.46.png$

More precisely, if $B$ lies between $A$ and $C$, then the arc $ABC$ is defined as the line segment $[AC]$. If $B'$ lies on the extension of $[AC]$, then the arc $AB'C$ is defined as a union of disjoint half-lines $[AX)$ and $[CY)$ in $(AC)$. In this case the arcs $ABC$ and $AB'C$ are called opposite to each other.

$截屏2021-02-19 上午9.17.46.png$

In addition, any half-line $[AB)$ will be regarded as an arc. If $A$ lies between $B$ and $X$, then $[AX)$ will be called oppostie to $[AB)$. This degenerate arc has only one endpoint $A$.

It will be convenient to use the notion of circline, that means circle or line. For example any arc is a subset of a circline; we also may use the term circline arc if we want to emphasise that the arc might be degenerate. Note that for any three distinct points $A, B$, and $C$ there is a unique circline arc $ABC$.

The following statement summarizes the discussion above.

Proposition $\PageIndex{1}$

Let $ABC$ be a circline arc and $X$ be a point distinct from $A$ and $C$. Then

(a) $X$ lies on the arc $ABC$ if and only if

$\measuredangle AXC = \measuredangle ABC;$

(b) $X$ lies on the arc opposite to $ABC$ if and only if

$\measuredangle AXC \equiv \measuredangle ABC + \pi$;

Exercise $\PageIndex{1}$

Given an acute triangle $ABC$ make a compass-and-ruler construction of the point $Z$ such that

$\measuredangle AZB = \measuredangle BZC = \measuredangle CZA = \pm \dfrac{2}{3} \cdot \pi$

Hint

$截屏2021-02-19 下午2.40.25.png$

Guess a construction from the diagram. To show that it produces the needed point, apply Theorem 9.2.1.

Exercise $\PageIndex{2}$

Suppose that point $P$ lies on the circumcircle of an equilateral triangle $ABC$ and $PA \le PB \le PC$. Show that $PA + PB = PC$.

Hint

$截屏2021-02-19 上午9.36.37.png$

Show that $P$ lies on the arc opposite from $ACB$; conclude that $\measuredangle APC = \measuredangle CPB = \pm \dfrac{\pi}{3}$.

Choose a point $A' \in [PC]$ such that $PA' = PA$. Note that $\triangle APA'$ is equilateral. Prove and use that $\triangle AA'C \cong \triangle APB.$

A quadrangle $ABCD$ is inscribed if all the points $A, B, C$, and $D$ lie on a circline $\Gamma$. If the arcs $ABC$ and $ADC$ are opposite, then we say that the points $A, B, C$, and $D$ appear on $\Gamma$ in the same cyclic order.

This definition makes it possible to formulate the following refinement of Corollary 9.3.2 which includes the degenerate quadrangles. It follows directly from Proposition $\PageIndex{1}$.

Proposition $\PageIndex{2}$

A quadrangle $ABCD$ is inscribed in a circline if and only if

$\measuredangle ABC = \measuredangle ADC$ or $\measuredangle ABC \equiv \measuredangle ADC + \pi.$

Moreover, the second identity holds if and only if the points $A,B,C,D$ appear on the circline in the same cyclic order.

Proposition \(\PageIndex{1}\)

Exercise \(\PageIndex{1}\)

Exercise \(\PageIndex{2}\)

Proposition \(\PageIndex{2}\)