Inscribed Angle: an angle made from points sitting on the circle's circumference. inscribed So in triangle BXC we know Angle BXC = 85°, and Angle XCB = 32°. Most geometry so far has involved triangles and quadrilaterals, which are formed by intervals on lines, and we turn now to the geometry of circles. Lines and. Circles have degrees. Triangles have degrees. When you see a simple 2 to 1 ratio you know somethings up. Here the vertices of our triangle are forced.

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Polygons which are equiangular and have their corresponding sides proportional are said to be similar. If also their corresponding sides are parallel, they are said to be similarly situated or homothetic Theorem 1 The triangles and circles of the areas of similar triangles or triangles and circles is equal to the ratio of the squares on corresponding sides.

The last equality follows from the fact that the triangles ABC and PQR are similar If Polygons are similar they can be divided up into the same number of similar triangles and it follows that the ratio of the areas of similar polygons is equal to the ratio of the squares on corresponding sides.

## Circle Theorems

This proves that triangles and circles line is a tangent, because it meets the circle only at T. It also proves that every point onexcept for T, lies outside the circle.

It remains to prove part b, that there is no other tangent to the circle at T. Let t be a tangent at Triangles and circles, and suppose, by way of contradiction, that t were not perpendicular to OT.

## Triangles and Circles - Pure - Geometry - Maths Reference with Worked Examples

Hence U also lies on the circle, contradicting the fact that t is a tangent. Tangents from an external point have equal length It is also a simple consequence of the triangles and circles theorem that the two tangents PT and PU have equal length.

Tangents to a circle from an external point have equal length. Tangents and trigonometry The right angle formed by a radius and tangent gives further opportunities for simple trigonometry.

### Triangles and Circles

Quadrilaterals with incircles The following exercise involves quadrilaterals within which an incircle can be triangles and circles tangent to all four sides.

Angles Subtended on the Same Arc Angles formed from two points on the circumference are equal to other angles, in the same arc, formed from those two points. Angle in a Semi-Circle Angles formed by drawing lines triangles and circles the ends of the diameter of a circle to its circumference form a right angle.

• Triangles in Circles :
• Circle Geometry
• Circle Theorems - Mathematics GCSE Revision
• Triangles in Circles
• Statements Of Some Theorems On The Circle.

So c is a right angle. Proof We can split the triangle in two by drawing a line from the centre of the circle to the point on the circumference our triangle touches.

We know that each of the lines which is a radius of the circle the green lines are triangles and circles same length.

Therefore each of the two triangles is isosceles and has a pair of equal angles.