CIRCLES FROM TRIANGLE

From any given triangle the following kind of cirlces are obtained.

Circumcircle       :            The circle which passes through all the vertices of a
triangle is known as a circumcircle.

In the given figure, PQR is a triangle, perpendicular bisectors of whose sides intersect each other at C (Circumcentre). If we draw a circle taking C as a centre and CQ=CP=CR as radius we will get a circle known as Circumcircle.

Incircle                 :               The circle which touches all the sides of a triangle is
known as incircle.

In the given figure let PQR is a given triangle and its angle bisectors intersect each other at I (Incen
tre).
Draw IM perpendicular to QR, the circle with centre I and radius IM is known as incircle of a triangle.

Excircle                :               The circle which touches the one side of a triangle and other two sides when extended is known as an excircle.

Let PQR be a given triangle, PQ and PR are extended Exterior angle bisectors of Q and R intersects at E₁ (Ex-centre). E1M is perpendicular drawn from E1 to QR. A circle with centre E1 and radius M is known as ex-circle of the triangle. In a given triangle we can draw three ex-circles as shown in the figure.

Nine Point Circle:            The circle which passes through

1. mid-point of each side,
2. foot of altitudes and
3. mid-point of line-segment joining vertex and the orthocentre.

Is known as a nine point circle and the centre of that circle is known as nine point centre.

In the given figure ABC is a triangle. D, E and F are the mid-points of sides of a triangle, L, M and N are the foot of perpendiculars from vertices to the opposite sides, H is the orthocenter, P, Q and R are the mid-points of line segment joining vertices and the orthocenter.

The circle which passes through D, E, F, L, M, N, P, Q and R is known as nine point circle.