Orthocentre and the Pedal Triangle:
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Orthocentre is a point of concurrence of altitude from the vertices to the opposite side of the triangle. The triangle LMN which is formed by joining feet of these 3 altitudes is called as pedal triangle.
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Remarks:
- Orthocentre of triangle is incentre of pedal triangle.
- If I1, I2 and I3 be centres of escribed circles which are opposite to A, B and C respectively and I centre of incircle then triangle ABC is pedal triangle of triangle I1 I2 I3 and I is orthocentre of triangle I1 I2I3 .
- The centroid of triangle lies on line joining circumcentre to orthocentre and divides it in ratio of 1 : 2
- Circle circumscribing the pedal triangle of the given triangle bisects the sides of given triangle and also the lines joining the vertices of the given triangle to the orthocentre of the given triangle. This circle is known as nine point circle.
- Circum centre of pedal triangle of a given triangle bisects line joining the circum centre of triangle to orthocentre.
Cyclic Quadrilateral:
A cyclic Quadrilateral is quadrilateral all the which vertices lies on a circle.
Note:
- Sum of opposite angles of the cyclic quadrilateral is 180°
- In the cyclic quadrilateral sum of products of the opposite sides is equal to the product of the diagonals. This is called as Ptolemy's theorem.
- If sum of opposite sides of the quadrilateral is equal, then and only then a circle can be inscribed in quadrilateral.
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