In any triangle ABC,

The triangle formed by joining the points F, E, and D is called the pedal triangle. In any triangle ABC,

a). The distance of the orthocentre from the vertices of the triangle ABC is-

Distance from $A = 2RcosA$

Distance from $B = 2RcosB$

Distance from $C = 2RcosC$

b). Distance of the orthocentre from the sides of the triangle ABC is-

$2RcosBcosC$ , $2RcosCcosA$ , $2RcosAcosB$

POINTS TO BE NOTED

1). The circumradius of a padel triangle is  $$\frac{R}{2}$$

2). Area of a pedal triangle is $$ 2\Delta cosA.cosB.cosC = \frac{R^2.sin2A.sin2B.sin2C}{2}$$

3). Length of the median AD, BE, and CF of the ∆ABC. is given by-

AD = $$\frac{\sqrt{(b^2+c^2+2bc.cosA)}}{2} =\frac{\sqrt{(2b^2+2c^2-a^2)}}{2}$$

BE = $$\frac{\sqrt{(a^2+c^2+2ac.cosB)}}{2} = \frac{\sqrt{(2a^2+2c^2-b^2)}}{2}$$

CF = $$\frac{\sqrt{(b^2+a^2+2ba.cosC)}}{2} =\frac{\sqrt{(2b^2+2a^2-c^2)}}{2}$$

4). Length of the angle bisector through vertex A= $$\frac{2bc.cos\left(\frac{A}{2}\right)}{(b+c)}$$

 Through vertex B =$$\frac{2ac.cos\left(\frac{B}{2}\right)}{(a+c)}$$

 Through vertex C = $$\frac{2ab.cos\left(\frac{C}{2}\right)}{(a+b)}$$

5). Length of the altitude from;

  From vertex A = $$\frac{a}{cotB+cotC}$$

  From vertex B = $$\frac{b}{cotC+cotA}$$

  From vertex C = $$\frac{c}{cotA+cotB}$$

6). Distance between circumcentre and the orthocentre = $$R\sqrt{(1-8cosA.cosB.cosC)}$$

7). Distance between circumcentre and incentre is=

$$\sqrt{(R^2-2Rr)}=R\sqrt{\left(1-8cos\left(\frac{A}{2}\right).cos\left(\frac{B}{2}\right).cos\left(\frac{C}{2}\right)\right)}$$

8). Distance between incentre and the orthocentre = $$\sqrt{(2r^2-4R^2.cosA.cosB.cosC)}$$

9). Distance between circumcentre and centroid =$$R^2-\frac{(a^2+b^2+c^2)}{9}$$

10). If circumcentre, centroid, and orthocentre are collinear and G divides OO’ in the ratio of 1:2

11). If I1, I2, and I3 are the center of the escribed circles opposite to the angle A, B, and C respectively.  And O is the orthocentre. Then

$$OI_1 =R\sqrt{\left(1+8sin\left(\frac{A}{2}\right).cos\left(\frac{B}{2}\right).cos\left(\frac{C}{2}\right)\right)}$$

$$OI_2 = R\sqrt{\left(1+8cos\left(\frac{A}{2}\right).sin\left(\frac{B}{2}\right).cos\left(\frac{C}{2}\right)\right)}$$

$$OI_3 = R\sqrt{\left(1+8cos\left(\frac{A}{2}\right).cos\left(\frac{B}{2}\right).sin\left(\frac{C}{2}\right)\right)}$$

CYCLIC QUADRILATERAL

POINTS TO BE NOTED

1). Area of the cyclic quadrilateral ABCD is = $$\sqrt{s(s-a)(s-b)(s-c)(s-d)}$$

2). Circumradius of the cyclic quadrilateral =

   $$\sqrt{\frac{(ab+cd)(ad+bc)(ac+bd)}{16(s-a)(s-b)(s-c)(s-d)}}$$

3). CosB = $\frac{(a^2+b^2-c^2-d^2)}{2(ab+cd)}$ and similarly other angles.

PTOLEMY THEOREM –

If ABCD is a cyclic quadrilateral then AC.BD = AB.CD+BC.AD

POINTS TO BE NOTED

1). If a regular polygon has n sides, then the sum of its internal angle is (n-2)π and each angle is $$\frac{\pi(n-2)}{n}$$

2). In a regular polygon the centroid, the circumcentre, and incentre are the same.

3). Area of the regular polygon = $$\frac{na^2.cot\left(\frac{\pi}{n}\right)}{4}$$

 $$ =\frac{ nR^2.sin\left(\frac{2\pi}{n}\right)}{2} = nr^2.tan\left(\frac{\pi}{n}\right)$$

Where n is the no of sides of the regular polygon, R is the radius of circumscribing circle, and r is the radius of the incircle of a polygon.

4). Radius of circumscribing circle =$$ R = \frac{a}{2sin\left(\frac{\pi}{n}\right)}= \frac{a}{2csc\left(\frac{\pi}{n}\right)}$$

Where a is the length of the side of the regular polygon of n sides.

5). Radius of inscribed circles r= $$\frac{a}{2cot\left(\frac{\pi}{n}\right)}$$