Tag Archives: circle geometry

March 18, 2025 · 6:04 am

Intersecting Secant Theorem

$CD$ is a tangent to the circle.

Prove $c^2=a(a+b)$

I am going to add two chords to the circle

Chord $AD$ and $BD$ are added

$\angle{BDC}=\angle{CAD}$ (angles in alternate segments are congruent)

$\angle{BCD}=\angle{DCA}$ (shared angle)

$\therefore \Delta BDC\cong \Delta{DAC}$ (AA)

Hence

$\frac{DC}{AC}=\frac{BC}{DC}$ (Corresponding sides in similar triangles)

$\frac{c}{a+b}=\frac{a}{c}$

$\therefore c^2=a(a+b)$

Filed under Circle Theorems, Geometry, Year 11 Specialist Mathematics

Tagged as circle geometry, Intersecting Secants Theorem

September 7, 2024 · 9:31 am

Geometry Problem

This problem is from Geometry Snacks by Ed Southall and Vincent Pantaloni – it’s a great book.

Two squares are constructed such that three vertices are collinear as shown. Find the value of the marked angle.

I started by marking in the right angles. And I added the diagonal of the larger square (pink line).

Because there are right angles at $O$ and $P$ , we know there is a circle, which has the diagonal of the square as its diameter (see second image below).

$\angle{RSP}$ is $45^{\circ}$ (Angle between the diagonal and side of a square)

$PORS$ is a cyclic quadrilateral.

In cyclic quadrilaterals opposite angles are supplementary.

Hence, $\angle{ROP}=180^{\circ}-45^{\circ}=135^{\circ}$

As $\angle{ROS}=90^{\circ}$ , $\angle{SOP}$ must be $45^{\circ}$

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Filed under Finding an angle, Geometry

Tagged as circle geometry, finding an angle, geometry, squares