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GeometryQuestion and Answers: Page 87

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Prove that if the lengths of a triangle form an arithmetic progression, then the centre of incircle and the centroid of triangle lie on a line parallel to the side of middle length of the triangle.

$${Prove}\:{that}\:{if}\:{the}\:{lengths}\:{of}\:{a}\: \\ $$$${triangle}\:{form}\:{an}\:{arithmetic} \\ $$$${progression},\:{then}\:{the}\:{centre}\:{of} \\ $$$${incircle}\:{and}\:{the}\:{centroid}\:{of} \\ $$$${triangle}\:{lie}\:{on}\:{a}\:{line}\:{parallel}\:{to} \\ $$$${the}\:{side}\:{of}\:{middle}\:{length}\:{of}\:{the} \\ $$$${triangle}. \\ $$

Question Number 62363    Answers: 2   Comments: 0

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Question Number 62121    Answers: 2   Comments: 0

A cube of unit edge length is held before a plane. Prove that the sum of the squares of the projected lengths of edges of the cube on the plane (irrespective of the orientation of the cube) is 8.

$${A}\:{cube}\:{of}\:{unit}\:{edge}\:{length}\: \\ $$$${is}\:{held}\:{before}\:{a}\:{plane}.\:{Prove}\:{that} \\ $$$${the}\:{sum}\:{of}\:{the}\:{squares}\:{of}\:{the} \\ $$$${projected}\:{lengths}\:{of}\:{edges}\:{of}\:{the}\:{cube} \\ $$$${on}\:{the}\:{plane}\:\left({irrespective}\:{of}\right. \\ $$$$\left.{the}\:{orientation}\:{of}\:{the}\:{cube}\right)\:{is}\:\mathrm{8}. \\ $$

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