Difference between revisions of "1960 IMO Problems/Problem 4"
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==Solution== | ==Solution== | ||
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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>. | 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 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>. | 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== | ||
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+ | https://youtu.be/M0_UdvxH890 | ||
==See Also== | ==See Also== |
Latest revision as of 01:43, 2 January 2023
Contents
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 |