Demonstrate the nature of the triangular median and trapezoidal median

It is known that in ABC, D and E are the midpoints of the edges AB and BC, respectively. Verification:

de=1 2bc Proof : d and e are the midpoints of the sides ab and bc respectively de is the median line, de bc ade abc ad ab=de bc ad=1 2ab de=1 2bc (it seems like this) Known: In trapezoidal ABCD, m and n are the midpoints of ab and cd Verification:

mn=1 2 (ab=cd) proof: extend an, the extension line of BC is o proof ADN ocd AD=OC, an=on n is the midpoint of AO, Mn is the median line of trapezoidal ABCD M, n are AB, Cd midpoint Mn is the median of triangular ABO mn=1 2bo Bo=BC+Co, CO=DA mn=1 2 (BC+AD).

The median line of the trapezoid is parallel to the two bases and is equal to half of the sum of the two bases.

Trapezoidal ABCD, E is the midpoint of AB, F is the midpoint of CD, connected EF.

Verification: EF is parallel to the two bases and equal to half of the sum of the two bases.

Ladder median line proof plot.

Proof: Connect the af and extend the extension line of the af to bc at o in adf and fco ad bc tong wu d= dco and dfa= cfo df df adf fco point e, f is ab, ao midpoint ef is the median line of the triangle ab ef ob i.e. ef bc ad bc bc ad (ef parallel two bottoms) ef is the median line of the triangle abo 2EF=ob ob=BC+co co=AD 2ef=bc+AD EF = half of AD + BC (EF is equal to half of the sum of the two bases), that is, the median line of the trapezoid is parallel to the upper spike and the lower two bottoms and is equal to half of the sum of the two bases.

In a triangle, the line segment that connects the midpoints of any two sides is called the median line of the triangle.

The median line of the triangle is parallel to the base edge and is half the length of the base edge.

The median line of the triangle is parallel to the third side of the triangle and is equal to half of the third side

It is known that in ABC, D and E are the midpoints of the sides AB and BC, respectively.

Verification: de=1 2bc

Proof: D and E are the midpoints of the edges AB and BC, respectively.

de is the median line, de bc

ade∽△abc

ad/ab=de/bc

ad=1/2ab

DE=1 2BC (which appears to be lead is the early form of the collapse) is known: in trapezoidal ABCD, M and N are the midpoints of AB and Cd.

Verification: mn=1 2(ab=cd).

Proof: Extend AN, the extension line of the delivery BC is O

Proof of ADN OCD

ad=oc,an=on

n is the ao-midpoint.

MN is the median line of trapezoidal ABCD.

m and n are the midpoints of ab and cd, respectively.

MN is the median line of the triangle ABO.

mn=1/2bo

bo=bc+co,co=da

mn=1/2(bc+ad)

The triangle median line is extrapolated from similar triangles and can be said to be a special case.

The median line of the trapezoid is the midpoint connecting the two waists of the trapezoid, and then extended, so that it forms a triangle with the extension line of the lower bottom, by proving congruence, the side of the upper bottom is equal to the distance of the extension of the lower bottom, and then this constitutes a triangle, and the median line of the triangle is proved above, and this conclusion is also used here to obtain that the median line of the trapezoid is parallel and equal to half of the sum of the upper and lower bases.

It is known that in ABC, D and E are the midpoints of the edges AB and BC, respectively.

Verification: de=1 2bc

Proof: D and E are the midpoints of the edges AB and BC, respectively.

de is the median line, de bc

ade∽△abc

ad/ab=de/bc

ad=1/2ab

de=1 2bc (looks like this).

It is known that in trapezoidal ABCD, M and N are the midpoints of AB and CD: mn=1 2(AB=CD).

Proof: Extend AN, the extension line of the delivery BC is O

Proof of ADN OCD

AD=OC and AN=ONN are the midpoints of AO.

MN is the median line of trapezoidal ABCD.

m and n are the midpoints of ab and cd, respectively.

MN is the median line of the triangle ABO.

mn=1/2bo

bo=bc+co,co=da

mn=1/2(bc+ad)

The trapezoidal median hall is parallel and equal to one-half of the upper bottom and lower bottom.

Trapezoidal ABCD, AB parallel CD

Connect to the Xun Zheng angle line AC intersection median line EF at G

According to the median line of the triangle (through the similar triangle can Chang Zen evidence, I don't know to ask me) can be known:

eg and =1 2cd, gf =1 2ab, so eg + gf and =1 2 (ab+cd).

Make a trapezoid, and then reverse the bottom upside down to form a parallelogram composed of the side and waist length of the upper and lower bottoms, and make the median line of the manuscript state, just right.

It is twice the length of a median line, and has the property of a parallelogram equal to half of the sum of the median line length of the upper bottom and the lower bottom.

From BC ef and BC=2ef, (similar triangular properties), AB and =2AE, AC and =2AF, then there is AE=EB, AF=Fc, so we can know that E, F is AB, the midpoint on AC, and EF BC, so EF is the median line of the triangle ABC.

The median line of the triangle has a dual relationship between position and quantity.

The median line of the triangle is parallel to the third side, and the median line of the triangle is equal to half of the third side.

It is usually widely used in calculation and proof problems, and it is necessary to understand and apply it proficiently.

Median line 1Median Concept:

1) Triangle median line definition: The line segment connecting the midpoints on both sides of the triangle is called the triangle median line

2) Definition of trapezoidal median line: The line segment connecting the midpoint of the two waists of the trapezoid is called the trapezoidal median line

Note: (1) Distinguish the median line of the triangle from the midline of the triangle The median line of the triangle is the line segment that connects a vertex with the midpoint of its opposite side, and the median line of the triangle is the line segment that connects the midpoints of the two sides of the triangle

2) The median line of the trapezoidal shape is the line segment that connects the midpoints of the two waists, not the line segment that connects the midpoints of the two bases

3) The connection between the two median line definitions: the triangle can be thought of as a trapezoid when the top and bottom are zero, and then the median line of the trapezoid becomes the median line of the triangle

2.Median line theorem:

1) Median line theorem of triangles: The median line of a triangle is parallel to the third side and equal to half of it

2) Trapezoidal median line theorem: The median line of the trapezoidal is parallel to the two bases and is equal to half of the sum of the two bases

The median line is an important line segment in triangles and trapezoids, and because of its properties, it is closely connected with the midpoint and parallel lines of the line segment, so it has a wide range of applications in the calculation and proof of geometric figures

parallel and equal to half of the third side.