9.3: Points on a common circle

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Recall that the diameter of a circle is a chord that passes thru the center. If $[XY]$ is the diameter of a circle with center $O$, then $\measuredangle XOY = \pi$. Hence Theorem 9.2.1 implies the following:

Corollary $\PageIndex{1}$

Suppose $\Gamma$ is a circle with the diameter $[AB]$. A triangle $ABC$ has right angle at $C$ if and only if $C \in \Gamma$.

Exercise $\PageIndex{1}$

Given four points $A, B, A'$, and $B'$, construct a point $Z$ such that both angles $AZB$ and $A'ZB'$ are right.

Hint: Construct the circles $\Gamma$ and $\Gamma'$ on the diameters $[AB]$ and $[A'B']$ respectively. By Corollary $\PageIndex{1}$, any point $Z$ in the intersection $\Gamma \cap \Gamma'$ will do.

Exercise $\PageIndex{2}$

Let $\triangle ABC$ be a nondegenerate triangle, $A'$ and $B'$ be foot points of altitudes from $A$ and $B$ respectfully. Show that the four points $A, B, A'$, and $B'$ lie on one circle. What is the center of this circle?

Hint

Note that $\measuredangle AA'B = \pm \dfrac{\pi}{2}$ and $\measuredangle AB'B = \pm \dfrac{\pi}{2}$. Then apply Corollary $\PageIndex{2}$ to $\square AA'BB'$.

If $O$ is the center of the circle, then $\measuredangle AOB \equiv 2 \cdot \measuredangle AA'B \equiv \pi$. That is, $O$ is the midpoint of $[AB]$.

Exercise $\PageIndex{3}$

Assume a line $\ell$, a circle with its center on $\ell$ and a point $P \not\in \ell$ are given. Make a ruler-only construction of the perpendicular to $\ell$ from $P$.

Hint

Guess the construction from the diagram. To prove it, apply Theorem 8.2.1 and Corollary $\PageIndex{1}$.

$截屏2021-02-18 下午2.34.09.png$

Exercise $\PageIndex{4}$

Suppose that lines $\ell$, $m$ and $n$ pass thru a point $P$; the lines $\ell$ and $m$ are tangent to a circle $\Gamma$ at $L$ and $M$; the line $n$ intersects $\Gamma$ at two points $X$ and $Y$. Let $N$ be the midpoint of $[XY]$. Show that the points $P, L, M$, and $N$ lie on one circle.

$截屏2021-02-18 下午2.13.51.png$

We say that a quadrangle $ABCD$ is inscribed in circle $\Gamma$ if all the points $A, B, C$, and $D$ lie on $\Gamma$.

Hint: Denote by $O$ the center of $\Gamma$. Use Corollary $\PageIndex{1}$ to show that the point lie on the circle with diameter $[PO]$.

Corollary $\PageIndex{2}$

A nondegenerate quadrangle $ABCD$ is inscribed in a circle if and only if

$2 \cdot \measuredangle ABC \equiv 2 \cdot \measuredangle ADC.$

Proof

Since $\square ABCD$ is nondegenerate, so is $\triangle ABC$. Let $O$ and $\Gamma$ denote the circulcenter and circumcircle of $\triangle ABC$ (they exist by Exercise 8.1.1).

$截屏2021-02-18 下午2.25.02.png$

According to Theorem 9.2.1,

$2 \cdot \measuredangle ABC \equiv \measuredangle AOC$.

From the same theorem, $D \ni \Gamma$ if and only if

$2 \cdot \measuredangle ADC \equiv \measuredangle AOC,$

hence the result.

Exercise $\PageIndex{5}$

Let $\Gamma$ and $\Gamma'$ be two circles that intersect at two distinct point: $A$ and $B$. Assume $[XY]$ and $[X'Y']$ are the chords of $\Gamma$ and $\Gamma'$ respectively, such that $A$ lies between $X$ and $X'$ and $B$ lies between $Y$ and $Y'$. Show that $(XY) \parallel (X'Y')$.

$截屏2021-02-18 下午2.32.08.png$

Hint: Apply Corollary $\PageIndex{2}$ twice for $\square ABYX$ and $\square ABY'X'$ and use the transversal property (Theorem 7.3.1).

Corollary \(\PageIndex{1}\)

Exercise \(\PageIndex{1}\)

Exercise \(\PageIndex{2}\)

Exercise \(\PageIndex{3}\)

Exercise \(\PageIndex{4}\)

Corollary \(\PageIndex{2}\)

Exercise \(\PageIndex{5}\)