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1.Given a regular tetrahedron ABCD with vertices A(0,0,0)B(a,0,0), C(0,a,0),and D(0,0,a).Calculate the volume V and the surface area S of this tetrahedron.

$$\mathrm{1}.\mathrm{Given}\:\mathrm{a}\:\mathrm{regular}\:\mathrm{tetrahedron}\:\boldsymbol{{ABCD}} \\ $$$$\mathrm{with}\:\mathrm{vertices}\:\boldsymbol{{A}}\left(\mathrm{0},\mathrm{0},\mathrm{0}\right)\boldsymbol{{B}}\left(\boldsymbol{{a}},\mathrm{0},\mathrm{0}\right), \\ $$$$\boldsymbol{{C}}\left(\mathrm{0},\boldsymbol{{a}},\mathrm{0}\right),\boldsymbol{\mathrm{and}}\:\boldsymbol{{D}}\left(\mathrm{0},\mathrm{0},\boldsymbol{{a}}\right).\mathrm{Calculate}\:\mathrm{the} \\ $$$$\:\mathrm{volume}\:\boldsymbol{{V}}\:\:\mathrm{and}\:\mathrm{the}\:\mathrm{surface}\:\mathrm{area}\:\boldsymbol{{S}}\:\boldsymbol{\mathrm{of}} \\ $$$$\mathrm{this}\:\mathrm{tetrahedron}. \\ $$

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Two circles are assumed that have the same center. How many circles are there that are tangent to both circles and pass through the assumed point?

$$ \\ $$Two circles are assumed that have the same center. How many circles are there that are tangent to both circles and pass through the assumed point?

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at what times, if exist, are the angles betwen the hour hand, the minute hand and the second hand of a clock exactly 120°? assume that the hands of the clock move uniformly.

$${at}\:{what}\:{times},\:{if}\:{exist},\:{are}\:{the}\: \\ $$$${angles}\:{betwen}\:{the}\:{hour}\:{hand},\:{the} \\ $$$${minute}\:{hand}\:{and}\:{the}\:{second}\:{hand} \\ $$$${of}\:{a}\:{clock}\:{exactly}\:\mathrm{120}°? \\ $$$${assume}\:{that}\:{the}\:{hands}\:{of}\:{the}\:{clock} \\ $$$${move}\:{uniformly}. \\ $$

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In a convex quadrilateral ABCD, diagonals AC and BD intersect at E, while perpendicular bisectors of AB and CD intersect at F, and those of BC and DA intersect at G. Prove: (1) E, F, and G are collinear, (2) AE:EC = BF:FD, and (3) CG:GD = AF:FB.

$$ \\ $$In a convex quadrilateral ABCD, diagonals AC and BD intersect at E, while perpendicular bisectors of AB and CD intersect at F, and those of BC and DA intersect at G. Prove: (1) E, F, and G are collinear, (2) AE:EC = BF:FD, and (3) CG:GD = AF:FB.

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