Mixing
Prove that triangle midline and median split themselves into halves
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How can I prove that median and midline in a triangle split themselves into halves?
Thanks a lot in advance!
geometry triangle
asked Jan 20 at 21:17
Xxx DddXxx Ddd
354
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add a comment |
$begingroup$
How can I prove that median and midline in a triangle split themselves into halves?
Thanks a lot in advance!
geometry triangle
asked Jan 20 at 21:17
Xxx DddXxx Ddd
354
$endgroup$
$begingroup$
Median I know, but what is "midline"?
$endgroup$
– DonAntonio
Jan 20 at 21:26
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@DonAntonio it connects centers of sides in a triangle
$endgroup$
– Xxx Ddd
Jan 20 at 21:30
$begingroup$
Ok, thanks. Do you know Thales theorems?
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– DonAntonio
Jan 20 at 22:01
$begingroup$
@DonAntonio yep, I know what that is
$endgroup$
– Xxx Ddd
Jan 20 at 22:03
add a comment |
$begingroup$
How can I prove that median and midline in a triangle split themselves into halves?
Thanks a lot in advance!
geometry triangle
asked Jan 20 at 21:17
Xxx DddXxx Ddd
354
$endgroup$
How can I prove that median and midline in a triangle split themselves into halves?
Thanks a lot in advance!
geometry triangle
geometry triangle
asked Jan 20 at 21:17
Xxx DddXxx Ddd
354
asked Jan 20 at 21:17
Xxx DddXxx Ddd
354
asked Jan 20 at 21:17
Xxx DddXxx Ddd
354
asked Jan 20 at 21:17
Xxx DddXxx Ddd
354
asked Jan 20 at 21:17
Xxx DddXxx Ddd
354
354
$begingroup$
Median I know, but what is "midline"?
$endgroup$
– DonAntonio
Jan 20 at 21:26
$begingroup$
@DonAntonio it connects centers of sides in a triangle
$endgroup$
– Xxx Ddd
Jan 20 at 21:30
$begingroup$
Ok, thanks. Do you know Thales theorems?
$endgroup$
– DonAntonio
Jan 20 at 22:01
$begingroup$
@DonAntonio yep, I know what that is
$endgroup$
– Xxx Ddd
Jan 20 at 22:03
add a comment |
$begingroup$
Median I know, but what is "midline"?
$endgroup$
– DonAntonio
Jan 20 at 21:26
$begingroup$
@DonAntonio it connects centers of sides in a triangle
$endgroup$
– Xxx Ddd
Jan 20 at 21:30
$begingroup$
Ok, thanks. Do you know Thales theorems?
$endgroup$
– DonAntonio
Jan 20 at 22:01
$begingroup$
@DonAntonio yep, I know what that is
$endgroup$
– Xxx Ddd
Jan 20 at 22:03
$begingroup$
Median I know, but what is "midline"?
$endgroup$
– DonAntonio
Jan 20 at 21:26
$begingroup$
Median I know, but what is "midline"?
$endgroup$
– DonAntonio
Jan 20 at 21:26
$begingroup$
@DonAntonio it connects centers of sides in a triangle
$endgroup$
– Xxx Ddd
Jan 20 at 21:30
$begingroup$
@DonAntonio it connects centers of sides in a triangle
$endgroup$
– Xxx Ddd
Jan 20 at 21:30
$begingroup$
Ok, thanks. Do you know Thales theorems?
$endgroup$
– DonAntonio
Jan 20 at 22:01
$begingroup$
Ok, thanks. Do you know Thales theorems?
$endgroup$
– DonAntonio
Jan 20 at 22:01
$begingroup$
@DonAntonio yep, I know what that is
$endgroup$
– Xxx Ddd
Jan 20 at 22:03
$begingroup$
@DonAntonio yep, I know what that is
$endgroup$
– Xxx Ddd
Jan 20 at 22:03
add a comment |
2 Answers
2
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oldest
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$begingroup$
HINT.
In a triangle a midline is parallel to the third side and is a half of that.
In a triangle a line through the midpoint of a side and parallel to another side, cuts the third side at its midpoint.
answered Jan 20 at 21:59
AretinoAretino
24.8k21444
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add a comment |
$begingroup$
Let $;Delta ABC;$ be the triangle, $;EF;$ the midline joining the middle points of $;AB,,AC;$ and $;AM;$ the median to $;BC;$, and let $;K;$ be the point of intersection of $;EF;$ and $;AM;$ .
Since $;EF||BC;$, Thales (the intercepts theorems for parallels) theorem tells us, applied twice, that
$$text{On};;Delta ABM;:;frac{AE}{EB}=frac{AK}{KM}=color{red}{frac{EK}{BM}};,;;;;;;;text{On};Delta AMC;:frac{AK}{KM}=frac{AF}{FC}=color{red}{frac{KF}{MC}} $$
Now just equate both red parts (which are equal!) and remember that $;BM=MC;$ ...
answered Jan 20 at 22:12
DonAntonioDonAntonio
179k1494232
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2 Answers
2
active
oldest
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2 Answers
2
active
oldest
votes
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$begingroup$
HINT.
In a triangle a midline is parallel to the third side and is a half of that.
In a triangle a line through the midpoint of a side and parallel to another side, cuts the third side at its midpoint.
answered Jan 20 at 21:59
AretinoAretino
24.8k21444
$endgroup$
add a comment |
$begingroup$
HINT.
In a triangle a midline is parallel to the third side and is a half of that.
In a triangle a line through the midpoint of a side and parallel to another side, cuts the third side at its midpoint.
answered Jan 20 at 21:59
AretinoAretino
24.8k21444
$endgroup$
add a comment |
$begingroup$
HINT.
In a triangle a midline is parallel to the third side and is a half of that.
In a triangle a line through the midpoint of a side and parallel to another side, cuts the third side at its midpoint.
answered Jan 20 at 21:59
AretinoAretino
24.8k21444
$endgroup$
HINT.
In a triangle a midline is parallel to the third side and is a half of that.
In a triangle a line through the midpoint of a side and parallel to another side, cuts the third side at its midpoint.
answered Jan 20 at 21:59
AretinoAretino
24.8k21444
answered Jan 20 at 21:59
AretinoAretino
24.8k21444
answered Jan 20 at 21:59
AretinoAretino
24.8k21444
answered Jan 20 at 21:59
AretinoAretino
24.8k21444
24.8k21444
add a comment |
add a comment |
$begingroup$
Let $;Delta ABC;$ be the triangle, $;EF;$ the midline joining the middle points of $;AB,,AC;$ and $;AM;$ the median to $;BC;$, and let $;K;$ be the point of intersection of $;EF;$ and $;AM;$ .
Since $;EF||BC;$, Thales (the intercepts theorems for parallels) theorem tells us, applied twice, that
$$text{On};;Delta ABM;:;frac{AE}{EB}=frac{AK}{KM}=color{red}{frac{EK}{BM}};,;;;;;;;text{On};Delta AMC;:frac{AK}{KM}=frac{AF}{FC}=color{red}{frac{KF}{MC}} $$
Now just equate both red parts (which are equal!) and remember that $;BM=MC;$ ...
answered Jan 20 at 22:12
DonAntonioDonAntonio
179k1494232
$endgroup$
add a comment |
$begingroup$
Let $;Delta ABC;$ be the triangle, $;EF;$ the midline joining the middle points of $;AB,,AC;$ and $;AM;$ the median to $;BC;$, and let $;K;$ be the point of intersection of $;EF;$ and $;AM;$ .
Since $;EF||BC;$, Thales (the intercepts theorems for parallels) theorem tells us, applied twice, that
$$text{On};;Delta ABM;:;frac{AE}{EB}=frac{AK}{KM}=color{red}{frac{EK}{BM}};,;;;;;;;text{On};Delta AMC;:frac{AK}{KM}=frac{AF}{FC}=color{red}{frac{KF}{MC}} $$
Now just equate both red parts (which are equal!) and remember that $;BM=MC;$ ...
answered Jan 20 at 22:12
DonAntonioDonAntonio
179k1494232
$endgroup$
add a comment |
$begingroup$
Let $;Delta ABC;$ be the triangle, $;EF;$ the midline joining the middle points of $;AB,,AC;$ and $;AM;$ the median to $;BC;$, and let $;K;$ be the point of intersection of $;EF;$ and $;AM;$ .
Since $;EF||BC;$, Thales (the intercepts theorems for parallels) theorem tells us, applied twice, that
$$text{On};;Delta ABM;:;frac{AE}{EB}=frac{AK}{KM}=color{red}{frac{EK}{BM}};,;;;;;;;text{On};Delta AMC;:frac{AK}{KM}=frac{AF}{FC}=color{red}{frac{KF}{MC}} $$
Now just equate both red parts (which are equal!) and remember that $;BM=MC;$ ...
answered Jan 20 at 22:12
DonAntonioDonAntonio
179k1494232
$endgroup$
Let $;Delta ABC;$ be the triangle, $;EF;$ the midline joining the middle points of $;AB,,AC;$ and $;AM;$ the median to $;BC;$, and let $;K;$ be the point of intersection of $;EF;$ and $;AM;$ .
Since $;EF||BC;$, Thales (the intercepts theorems for parallels) theorem tells us, applied twice, that
$$text{On};;Delta ABM;:;frac{AE}{EB}=frac{AK}{KM}=color{red}{frac{EK}{BM}};,;;;;;;;text{On};Delta AMC;:frac{AK}{KM}=frac{AF}{FC}=color{red}{frac{KF}{MC}} $$
Now just equate both red parts (which are equal!) and remember that $;BM=MC;$ ...
answered Jan 20 at 22:12
DonAntonioDonAntonio
179k1494232
answered Jan 20 at 22:12
DonAntonioDonAntonio
179k1494232
answered Jan 20 at 22:12
DonAntonioDonAntonio
179k1494232
answered Jan 20 at 22:12
DonAntonioDonAntonio
179k1494232
179k1494232
add a comment |
add a comment |
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$begingroup$
Median I know, but what is "midline"?
$endgroup$
– DonAntonio
Jan 20 at 21:26
$begingroup$
@DonAntonio it connects centers of sides in a triangle
$endgroup$
– Xxx Ddd
Jan 20 at 21:30
$begingroup$
Ok, thanks. Do you know Thales theorems?
$endgroup$
– DonAntonio
Jan 20 at 22:01
$begingroup$
@DonAntonio yep, I know what that is
$endgroup$
– Xxx Ddd
Jan 20 at 22:03