Three tangents are drawn at random to a given circle: Show that the odds are $3$ to $1$ against the circle being inscribed in the triangle formed by them.

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Solution

The incircle is the largest circle inscribed that fits in a triangle. The sides of the triangles are the tangents to this circle. Let us draw a circle and d 3 tangents to it at random as shown below.

Draw the parallel lines to each of the tangents so that these three lines are also tangents to the same circle as shown in the figure.
$T h e o d d s o f g e t t i n g t h i s t r i a n g l e = 6 / 2 = 3 / 1$
Hence, the odds are $3$ to $1$ against the circle being inscribed in the triangle formed by the three tangents.
Hence, proved.

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Three tangents are drawn at random to a given circle: Show that the odds are 3 to 1 against the circle being inscribed in the triangle formed by them.

Three tangents are drawn at random to a given circle: Show that the odds are $3$ to $1$ against the circle being inscribed in the triangle formed by them.