The area of the triangle whose vertices are the incentre, circumcentre and centroid of the triangle AOB in sq. units is

Engineering

Mathematics

Family of Straight lines

Question

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radius of the incircle,

$r = \frac{Δ}{s} = \frac{(18 \cdot 24) 2}{2 (18 + 24 + 30)} = \frac{18 \cdot 24}{72} = 6$

Hence coordinates of incentre (6, 6) ; orthocentre is (0, 0)

distance $OI = \sqrt{36 + 36} = 6 \sqrt{2}$

II–part:

$centroid is = (\frac{18}{3}, \frac{24}{3}) = (6, 8)$

Hence the triangle formed by I, G and circumcentre is

$∴ A = \frac{1}{2} |\begin{array}{ccc} 6 & 6 & 1 \\ 9 & 12 & 1 \\ 6 & 8 & 1 \end{array}| = \frac{1}{2} \cdot 2 \cdot 3 \cdot | \cdot | \begin{array}{ccc} 2 & 3 & 1 \\ 3 & 6 & 1 \\ 2 & 4 & 1 \end{array} | = 3 \cdot | \begin{array}{ccc} 2 & - 1 & 1 \\ 3 & 0 & 1 \\ 2 & 0 & 1 \end{array} ∣= 3 sq. units$

$Let \frac{1}{r_{1}}, \frac{1}{r_{2}}, \frac{1}{r_{3}} are in A.P.$

or $\frac{s - a}{Δ}, \frac{s - b}{Δ}, \frac{s - c}{Δ} in A.P.$

or s – a, s – b, s – c in A.P.
or a, b, c in A.P.
which is true for 18, 24, 30