Difference between revisions of "1960 IMO Problems/Problem 4"
(New page: ==Problem== ==Solution== {{solution}} ==See Also== {{IMO box|year=1960|num-b=3|num-a=5}}) |
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==Problem== | ==Problem== | ||
+ | Construct triangle <math>ABC</math>, given <math>h_a</math>, <math>h_b</math> (the altitudes from <math>A</math> and <math>B</math>), and <math>m_a</math>, the median from vertex <math>A</math>. | ||
==Solution== | ==Solution== | ||
− | {{ | + | Let <math>M_a</math>, <math>M_b</math>, and <math>M_c</math> be the midpoints of sides <math>\overline{BC}</math>, <math>\overline{CA}</math>, and <math>\overline{AB}</math>, respectively. Let <math>H_a</math>, <math>H_b</math>, and <math>H_c</math> be the feet of the altitudes from <math>A</math>, <math>B</math>, and <math>C</math> to their opposite sides, respectively. Since <math>\triangle ABC\sim\triangle M_bM_aC</math>, with <math>M_bM_a=\frac12 AB</math>, the distance from <math>M_a</math> to side <math>\overline{AC}</math> is <math>\frac{h_b}{2}</math>. |
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+ | Construct <math>AM_a</math> with length <math>m_a</math>. Draw a circle centered at <math>A</math> with radius <math>h_a</math>. Construct the tangent <math>l_1</math> to this circle through <math>M_a</math>. <math>\overline{BC}</math> lies on <math>l_1</math>. | ||
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+ | Draw a circle centered at <math>M_a</math> with radius <math>\frac{h_b}{2}</math>. Construct the tangent <math>l_2</math> to this circle through <math>A</math>. <math>\overline{AC}</math> lies on <math>l_2</math>. Then <math>C=l_1\cap l_2</math>. | ||
+ | |||
+ | Construct the line <math>l_3</math> parallel to <math>l_2</math> so that the distance between <math>l_2</math> and <math>l_3</math> is <math>h_b</math> and <math>M_a</math> lies between these lines. <math>B</math> lies on <math>l_3</math>. Then <math>B=l_1\cap l_3</math>. | ||
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+ | ==Video Solution== | ||
+ | |||
+ | https://youtu.be/M0_UdvxH890 | ||
==See Also== | ==See Also== | ||
− | {{ | + | {{IMO7 box|year=1960|num-b=3|num-a=5}} |
+ | [[Category:Olympiad Geometry Problems]] | ||
+ | [[Category:Geometric Construction Problems]] |
Latest revision as of 00:43, 2 January 2023
Contents
[hide]Problem
Construct triangle , given , (the altitudes from and ), and , the median from vertex .
Solution
Let , , and be the midpoints of sides , , and , respectively. Let , , and be the feet of the altitudes from , , and to their opposite sides, respectively. Since , with , the distance from to side is .
Construct with length . Draw a circle centered at with radius . Construct the tangent to this circle through . lies on .
Draw a circle centered at with radius . Construct the tangent to this circle through . lies on . Then .
Construct the line parallel to so that the distance between and is and lies between these lines. lies on . Then .
Video Solution
See Also
1960 IMO (Problems) | ||
Preceded by Problem 3 |
1 • 2 • 3 • 4 • 5 • 6 • 7 | Followed by Problem 5 |