# Circle Geometry

• Oct 2nd 2008, 05:00 PM
xwrathbringerx
Circle Geometry
From an external point T, tangents are drawn to a circle with centre O, touching the circle at P and Q. Angle PTQ is acute. The diameter PB produced meets the tangent TQ at A. Let x=angle AQB.

(i) Prove that angle PTQ=2x

(ii) Prove that triangles PBQ and TOQ are similar.

(iii) Hence show that BQ*OT=2(OP)^2

• Oct 2nd 2008, 05:53 PM
Nacho
$
\measuredangle OQB = 2x \Rightarrow \measuredangle QOP = 180 - 2x{\text{ and }}\vartriangle OTQ \cong \vartriangle OTP \Rightarrow
$

$
\begin{gathered}
\measuredangle TOC = \measuredangle TOP = 90 - x \hfill \\
\Rightarrow \measuredangle OTC = \measuredangle OTP = x \Leftrightarrow \measuredangle PTQ = 2x \hfill \\
\end{gathered}
$

$
{\text{The }}\vartriangle QOB{\text{ have 2 sizes same}}{\text{, because }}\overline {BQ} \wedge \overline {BO} {\text{ are radius}}
$

$
\begin{gathered}
\Rightarrow \measuredangle OBQ = 90 - x, \hfill \\
{\text{look that }}\measuredangle BQP = 90 \hfill \\
\end{gathered}
$

$
{\text{because }}\overline {BP} {\text{ is diameter and how I prove the angles of }}\vartriangle OTQ
$

$
\Rightarrow \vartriangle TOQ \sim \vartriangle PBQ
$

now use de similar to show the iii)

is very hard write in latex in this forum (Headbang)
• Oct 4th 2008, 06:45 AM
bjhopper
circle geometry
originally posted by xwrathbringerx

Clues for you

angle BQP =90 deg halfarc of diameter
angle OQT =90 drg tangent radius property
angle x = half arc BQ tangent chord property
angle BPQ = half arc BQ angle between chords at circle property
angle BPQ = x
use the property of angle plus complement = 90 deg to define additional angles

bj