# inscribed triangle problem

#### jimi

'ABC is an acute-angled triangle inscribed in a circle and P, Q, R are the midpoints of the minor arcs BC, CA, AB respectively.
Prove that AP is perpendicular to QR.'
I know that the lines from the midpoints to the centre of the circle are perpendicular bisectors of the sides of the triangle
but I'm struggling to know how to use this.

#### Idea

AP and QR meet at M

measure of angle PMQ =

$$\displaystyle \frac{\text{arc} \text{PQ} + \text{arc} \text{AR} }{2}=\frac{\text{arc} \text{PC} + \text{arc} \text{CQ} + \text{arc} \text{AR}}{2}=90 \text{degrees}$$

#### jimi

Idea - thank you for responding but I don't understand your answer. When you write arcPQ, do you mean the angle subtended at the centre of the circle by arc PQ?

#### Plato

MHF Helper
Idea - thank you for responding but I don't understand your answer. When you write arcPQ, do you mean the angle subtended at the centre of the circle by arc PQ?
$m(\angle ABC) = \frac{1}{2}m\left( {arcAC} \right)$ the inscribed angle theorem.

Note that $m\left( {arcAQ}= \right)m\left( {arcQC} \right)=\frac{1}{2}m\left( {arcAC} \right).$

Use the intersecting chord theorem to show that $m(\angle AGQ) = \dfrac{\pi }{2}$, where $\left\{ G \right\} = \overline {AP} \cap \overline {RQ}$

#### Idea

central angle is twice inscribed angle so $$\displaystyle ROP=2RQP$$ where $$\displaystyle O$$ is the center of the circle

similarly $$\displaystyle AOQ=2APQ$$

$$\displaystyle 4*RQP + 4*APQ = 2*ROP + 2*AOQ = 360$$ because this is once around the circle

therefore $$\displaystyle RQP + APQ = 90$$

$$\displaystyle MQP + MPQ = 90$$ in triangle $$\displaystyle MPQ$$

angle $$\displaystyle M=90$$

#### Plato

MHF Helper

central angle is twice inscribed angle so $$\displaystyle ROP=2RQP$$ where $$\displaystyle O$$ is the center of the circle

similarly $$\displaystyle AOQ=2APQ$$

$$\displaystyle 4*RQP + 4*APQ = 2*ROP + 2*AOQ = 360$$ because this is once around the circle

therefore $$\displaystyle RQP + APQ = 90$$

$$\displaystyle MQP + MPQ = 90$$ in triangle $$\displaystyle MPQ$$

angle $$\displaystyle \color{red}M=90$$
Is that the number ninety or some ninety mystery units like degrees?

1 person

90 degrees