# Heights and Distances

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1.
a = h(cot α - cot β) = \tt \frac{h \ sin\left(\beta-\alpha\right)}{sin \alpha.sin\beta}
∴ h = a sin α sin β cosec (β - α) and
d = h cot β = α sin α cos β. cosec (β - α)

2.
H = x cot α tan (α + β)

3.
a = h (cot α + cot β). where by
h = a sin α. sin β. cosec (α + β) and
d = cot β = a sin α. cos β. cosec (α + β)

4.
\tt H=\frac{hcot \beta}{cot\alpha}

5.
\tt h=\frac{Hsin \left(\beta-\alpha\right)}{cos\alpha \ sin\beta}\ or \ H = \frac{h \ cot \ \alpha}{cot\alpha- \ cot\beta}

6.
\tt H=\frac{a\ sin\left(\alpha+\beta\right)}{sin\left(\beta-\alpha\right)}

7.
AB = CD. Then, x = y \tt tan\left(\frac{\alpha+\beta}{2}\right)

8.
In any triangle ABC if BD : DC = m : n and \tt \angle BAD =\alpha \ and \ \angle CAD=\beta \ and \ \angle ADC=\theta \ then \left(m + n\right)cot \theta = m \ cot \alpha-n \ cot \beta.