# PROPERTIES AND SOLUTIONS OF TRIANGLE

Here, there are some key concepts regarding the properties and solutions of triangles. these are very important for the students who preparing for IIT-JEE and other equivalent exams.

## POINTS TO BE REMEMBERED

### LAWS OF SINE OR SINE RULE

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The sides of a triangle are proportional to the sines of opposite angles i.e, In an ∆ABC,   $$\frac{a}{sinA} = \frac{b}{sinB} = \frac{c}{sinC} = k$$
Where k is some constant.

### LAWS OF COSINES OR COSINES RULE–

In any ∆ABC , We have

\begin{align*}
CosA& = \frac{\left(b^2 +c^2 – a^2\right)}{2bc}\\
CosB& = \frac{\left(c^2 + a^2 – b^2\right)}{2ac}\\
CosC &= \frac{\left( a^2 + b^2 – c^2\right)}{2ab}
\end{align*}

### PROJECTION FORMULA –

In any ∆ABC
\begin{align*}
a& = b.cosC + c.cosB\\
b& = a.cosC + c.cosA\\
c& = a.cosB + b.cosA
\end{align*}

### LAWS OF TANGENT OR TANGENT RULES (Napier’s Analogy)

\begin{align*}
tan\left(\frac{B-C}{2}\right)&= \left(\frac{b-c}{b+c}\right)cot\left(\frac{A}{2}\right)\\
tan\left(\frac{A-B}{2}\right)& = \left(\frac{a-b}{a+b}\right)cot\left(\frac{C}{2}\right)\\
tan\left(\frac{C-A}{2}\right)& = \left(\frac{c-a}{c+a}\right)cot\left(\frac{B}{2}\right)
\end{align*}

### HALF ANGLE FORMULAE OR SEMI-SUM FORMULAE

\begin{align*}
sin\left(\frac{A}{2}\right)& = \sqrt{\frac{(s-b)(s-c)}{bc}}\\
sin\left(\frac{B}{2}\right)& = \sqrt{\frac{(s-c)(s-a)}{ac}}\\
sin\left(\frac{C}{2}\right)& = \sqrt{\frac{(s-a)(s-b)}{ab}}
\end{align*}
\begin{align*}
cos\left(\frac{A}{2}\right)&= \sqrt{\frac{s(s-a)}{bc}}\\
cos\left(\frac{B}{2}\right)& = \sqrt{\frac{s(s-b)}{ac}}\\
cos\left(\frac{C}{2}\right)& = \sqrt{\frac{s(s-c)}{ab}}
\end{align*}
\begin{align*}
tan\left(\frac{A}{2}\right)& = \sqrt{\frac{(s-b)(s-c)}{s(s-a)}}\\
tan\left(\frac{B}{2}\right)& = \sqrt{\frac{(s-c)(s-a)}{s(s-b}}\\
tan\left(\frac{C}{2}\right)& = \sqrt{\frac{(s-a)(s-b)}{s(s-c)}}
\end{align*}

### AREA OF TRIANGLE –

The area of any triangle ABC can be given by:
$$\Delta = \frac{bc.sinA}{2} = \frac{ac.sinB}{2} = \frac{ab.sinC}{2}$$

### HERON’S FORMULA –

In a ∆ABC , if a+b+c = 2s, then it’s area can given by:
$$\Delta = \sqrt{s(s-a)(s-b)(s-c)}$$

### CIRCUMCIRCLE OF A TRIANGLE

In any triangle ABC,

a).$$R = \frac{a}{2sinA}= \frac{b}{2sinB} =\frac{ c}{2sinC}$$
b). $$R = \frac{abc}{4\Delta}$$
Everywhere  means area

### INCIRCLE OF A TRIANGLE

In any triangle ABC,
a).$$r = \frac{\Delta}{s}$$
b). $$r = (s-a)tan\left(\frac{A}{2}\right)= (s-b)tan\left(\frac{B}{2}\right)= (s-c)tan\left(\frac{C}{2}\right)$$
c).\begin{align*}
r& = \frac{a.sin\left(\frac{B}{2}\right).sin\left(\frac{C}{2}\right)}{cos\left(\frac{A}{2}\right)}\\
r &= \frac{b.sin\left(\frac{A}{2}\right).sin\left(\frac{C}{2}\right)}{cos\left(\frac{B}{2}\right)}\\
r& = \frac{c.sin\left(\frac{B}{2}\right).sin\left(\frac{A}{2}\right)}{cos\left(\frac{C}{2}\right)}
\end{align*}
d).$$r= 4Rsin\left(\frac{A}{2}\right).sin\left(\frac{B}{2}\right).sin\left(\frac{C}{2}\right)$$

### EXCRIBED CIRCLE OF A TRIANGLE

In any triangle ABC
a).
\begin{align*}
r_1& =\frac{\Delta}{s-a}\\
r_2&= \frac{\Delta}{s-b}\\
r_3&= \frac{\Delta}{s-c}
\end{align*}
b).
\begin{align*}
r1& = s.tan\left(\frac{A}{2}\right)\\
r2& = s.tan\left(\frac{B}{2}\right)\\
r3& = s.tan\left(\frac{C}{2}\right)
\end{align*}
c).
\begin{align*}
r_1 = \frac{a.cos\left(\frac{B}{2}\right).cos\left(\frac{C}{2}\right)}{cos\left(\frac{A}{2}\right)}\\
r_2 = \frac{b.cos\left(\frac{C}{2}\right).cos\left(\frac{A}{2}\right)}{cos\left(\frac{B}{2}\right)}\\
r_3 = \frac{c.cos\left(\frac{A}{2}\right).cos\left(\frac{B}{2}\right)}{cos\left(\frac{C}{2}\right)}
\end{align*}
d).
\begin{align*}
r_1& = 4R.sin\left(\frac{A}{2}\right).cos\left(\frac{B}{2}\right).cos\left(\frac{C}{2}\right)\\
r_2&= 4R.sin\left(\frac{B}{2}\right).cos\left(\frac{A}{2}\right).cos\left(\frac{C}{2}\right)\\
r_3&= 4R.sin\left(\frac{C}{2}\right).cos\left(\frac{A}{2}\right).cos\left(\frac{B}{2}\right)
\end{align*}

### ORTHOCENTRE OF A TRIANGLE

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

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

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

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

#### POINTS TO BE NOTED

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

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

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

### PTOLEMY THEOREM –

#### REGULAR POLYGON

##### 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= $$\frac{a}{2cot\left(\frac{\pi}{n}\right)}$$