Two. However, you can actually do it with just one. Consider the median AD of triangle ABC. Then the point G, 2/3 of the way from A to D, is the centroid. This process (2/3 of the way from the vertex to the opposite side) can be applied to any median.
Let the triangle be ABC and suppose the median AD is also an altitude.AD is a median, therefore BD = CDAD is an altitude, therefore angle ADB = angle ADC = 90 degreesThen, in triangles ABD and ACD,AD is common,angle ADB = angle ADCand BD = CDTherefore the two triangles are congruent (SAS).And therefore AB = AC, that is, the triangle is isosceles.
No.
It depends on whether you mean median in terms of statistical data or the line joining a vertex of a triangle to the midpoint of the opposite side. The Statistical Median: If there are n observations where n is odd, then the median is the (n+1)/2 smallest observation. If n is even, then the median is the arithmetical mean of the n/2 and n/2+1 smallest observations. The Coordinate Geometry Median: Given triangle ABC, with A = (xa, ya), B = (xb, yb) and C = (xc, yc), the midpoint, D of BC is [(xb+xc)/2, (yb+yc)/2] and so, the equation of the median AD is (y-ya)/(x-xa) = (2ya-yb-yc)/(2xa-xb-xc)
Any side: you can select the order in which the vertices are named!
Isosceles
It is isosceles.
It is isosceles.
It is a median.
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m^2=(2b^2+2c^2-a^2)/4 where m is the median of triangle ABC.
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Two. However, you can actually do it with just one. Consider the median AD of triangle ABC. Then the point G, 2/3 of the way from A to D, is the centroid. This process (2/3 of the way from the vertex to the opposite side) can be applied to any median.
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Let the triangle be ABC and suppose the median AD is also an altitude.AD is a median, therefore BD = CDAD is an altitude, therefore angle ADB = angle ADC = 90 degreesThen, in triangles ABD and ACD,AD is common,angle ADB = angle ADCand BD = CDTherefore the two triangles are congruent (SAS).And therefore AB = AC, that is, the triangle is isosceles.