Triangle $ADP$ and $AEN$ has the same area [closed]
$begingroup$
Let consider triangle $ABC$ and the circumcircle of $ABC$. Let $D$ and $E$ on $(BC)$ such that the angle $BAD$ and $ CAE$ has the same measure. Consider $S$ and $T$ on the circle such that the segments $AS$ and $AT$ has the same length and $P$ and $N$ be the intersection of $ES$ and $DT$ with $AB$ and $AC$. Then the triangle $ADP$ and $AEN$ has the same area.
geometry
asked Dec 6 '18 at 19:11
rafarafa
598212
$endgroup$
closed as off-topic by GNUSupporter 8964民主女神 地下教會, KReiser, José Carlos Santos, Rebellos, Cesareo Dec 7 '18 at 11:54
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – GNUSupporter 8964民主女神 地下教會, KReiser, José Carlos Santos, Rebellos, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
Let consider triangle $ABC$ and the circumcircle of $ABC$. Let $D$ and $E$ on $(BC)$ such that the angle $BAD$ and $ CAE$ has the same measure. Consider $S$ and $T$ on the circle such that the segments $AS$ and $AT$ has the same length and $P$ and $N$ be the intersection of $ES$ and $DT$ with $AB$ and $AC$. Then the triangle $ADP$ and $AEN$ has the same area.
geometry
asked Dec 6 '18 at 19:11
rafarafa
598212
$endgroup$
closed as off-topic by GNUSupporter 8964民主女神 地下教會, KReiser, José Carlos Santos, Rebellos, Cesareo Dec 7 '18 at 11:54
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – GNUSupporter 8964民主女神 地下教會, KReiser, José Carlos Santos, Rebellos, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.
$begingroup$
Please include a diagram. [and some work you did]
$endgroup$
– coffeemath
Dec 6 '18 at 19:17
$begingroup$
@coffeemath Actually I tried to apply the formula for area using sin.I don't know nothing about P and N. That is my problem
$endgroup$
– rafa
Dec 6 '18 at 19:22
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Still, a diagram showing just A,B,C,D,E would be nice.
$endgroup$
– coffeemath
Dec 6 '18 at 19:23
$begingroup$
@coffeemath I don't know how to use it
$endgroup$
– rafa
Dec 6 '18 at 19:47
add a comment |
$begingroup$
Let consider triangle $ABC$ and the circumcircle of $ABC$. Let $D$ and $E$ on $(BC)$ such that the angle $BAD$ and $ CAE$ has the same measure. Consider $S$ and $T$ on the circle such that the segments $AS$ and $AT$ has the same length and $P$ and $N$ be the intersection of $ES$ and $DT$ with $AB$ and $AC$. Then the triangle $ADP$ and $AEN$ has the same area.
geometry
asked Dec 6 '18 at 19:11
rafarafa
598212
$endgroup$
Let consider triangle $ABC$ and the circumcircle of $ABC$. Let $D$ and $E$ on $(BC)$ such that the angle $BAD$ and $ CAE$ has the same measure. Consider $S$ and $T$ on the circle such that the segments $AS$ and $AT$ has the same length and $P$ and $N$ be the intersection of $ES$ and $DT$ with $AB$ and $AC$. Then the triangle $ADP$ and $AEN$ has the same area.
geometry
geometry
asked Dec 6 '18 at 19:11
rafarafa
598212
asked Dec 6 '18 at 19:11
rafarafa
598212
asked Dec 6 '18 at 19:11
rafarafa
598212
asked Dec 6 '18 at 19:11
rafarafa
598212
asked Dec 6 '18 at 19:11
rafarafa
598212
598212
closed as off-topic by GNUSupporter 8964民主女神 地下教會, KReiser, José Carlos Santos, Rebellos, Cesareo Dec 7 '18 at 11:54
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – GNUSupporter 8964民主女神 地下教會, KReiser, José Carlos Santos, Rebellos, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by GNUSupporter 8964民主女神 地下教會, KReiser, José Carlos Santos, Rebellos, Cesareo Dec 7 '18 at 11:54
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – GNUSupporter 8964民主女神 地下教會, KReiser, José Carlos Santos, Rebellos, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.
$begingroup$
Please include a diagram. [and some work you did]
$endgroup$
– coffeemath
Dec 6 '18 at 19:17
$begingroup$
@coffeemath Actually I tried to apply the formula for area using sin.I don't know nothing about P and N. That is my problem
$endgroup$
– rafa
Dec 6 '18 at 19:22
$begingroup$
Still, a diagram showing just A,B,C,D,E would be nice.
$endgroup$
– coffeemath
Dec 6 '18 at 19:23
$begingroup$
@coffeemath I don't know how to use it
$endgroup$
– rafa
Dec 6 '18 at 19:47
add a comment |
$begingroup$
Please include a diagram. [and some work you did]
$endgroup$
– coffeemath
Dec 6 '18 at 19:17
$begingroup$
@coffeemath Actually I tried to apply the formula for area using sin.I don't know nothing about P and N. That is my problem
$endgroup$
– rafa
Dec 6 '18 at 19:22
$begingroup$
Still, a diagram showing just A,B,C,D,E would be nice.
$endgroup$
– coffeemath
Dec 6 '18 at 19:23
$begingroup$
@coffeemath I don't know how to use it
$endgroup$
– rafa
Dec 6 '18 at 19:47
$begingroup$
Please include a diagram. [and some work you did]
$endgroup$
– coffeemath
Dec 6 '18 at 19:17
$begingroup$
Please include a diagram. [and some work you did]
$endgroup$
– coffeemath
Dec 6 '18 at 19:17
$begingroup$
@coffeemath Actually I tried to apply the formula for area using sin.I don't know nothing about P and N. That is my problem
$endgroup$
– rafa
Dec 6 '18 at 19:22
$begingroup$
@coffeemath Actually I tried to apply the formula for area using sin.I don't know nothing about P and N. That is my problem
$endgroup$
– rafa
Dec 6 '18 at 19:22
$begingroup$
Still, a diagram showing just A,B,C,D,E would be nice.
$endgroup$
– coffeemath
Dec 6 '18 at 19:23
$begingroup$
Still, a diagram showing just A,B,C,D,E would be nice.
$endgroup$
– coffeemath
Dec 6 '18 at 19:23
$begingroup$
@coffeemath I don't know how to use it
$endgroup$
– rafa
Dec 6 '18 at 19:47
$begingroup$
@coffeemath I don't know how to use it
$endgroup$
– rafa
Dec 6 '18 at 19:47
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
As $AS=AT$, $angle AES=angle ADT$. As $angle BAD=angle CAE$, $angle PAE= angle NAD$.
$therefore triangle PAEsim triangle NAD$, whence, $frac{PA}{EA}=frac{AN}{AD}implies frac{1}{2}PA.ADsinangle{PAD}=frac{1}{2}EA.ANsinangle{EAN}$
Thus, $$Area(triangle PAD)=Area(triangle PAN)$$
$blacksquare$
answered Dec 7 '18 at 6:50
Anubhab GhosalAnubhab Ghosal
1,16919
$endgroup$
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
As $AS=AT$, $angle AES=angle ADT$. As $angle BAD=angle CAE$, $angle PAE= angle NAD$.
$therefore triangle PAEsim triangle NAD$, whence, $frac{PA}{EA}=frac{AN}{AD}implies frac{1}{2}PA.ADsinangle{PAD}=frac{1}{2}EA.ANsinangle{EAN}$
Thus, $$Area(triangle PAD)=Area(triangle PAN)$$
$blacksquare$
answered Dec 7 '18 at 6:50
Anubhab GhosalAnubhab Ghosal
1,16919
$endgroup$
add a comment |
$begingroup$
As $AS=AT$, $angle AES=angle ADT$. As $angle BAD=angle CAE$, $angle PAE= angle NAD$.
$therefore triangle PAEsim triangle NAD$, whence, $frac{PA}{EA}=frac{AN}{AD}implies frac{1}{2}PA.ADsinangle{PAD}=frac{1}{2}EA.ANsinangle{EAN}$
Thus, $$Area(triangle PAD)=Area(triangle PAN)$$
$blacksquare$
answered Dec 7 '18 at 6:50
Anubhab GhosalAnubhab Ghosal
1,16919
$endgroup$
add a comment |
$begingroup$
As $AS=AT$, $angle AES=angle ADT$. As $angle BAD=angle CAE$, $angle PAE= angle NAD$.
$therefore triangle PAEsim triangle NAD$, whence, $frac{PA}{EA}=frac{AN}{AD}implies frac{1}{2}PA.ADsinangle{PAD}=frac{1}{2}EA.ANsinangle{EAN}$
Thus, $$Area(triangle PAD)=Area(triangle PAN)$$
$blacksquare$
answered Dec 7 '18 at 6:50
Anubhab GhosalAnubhab Ghosal
1,16919
$endgroup$
As $AS=AT$, $angle AES=angle ADT$. As $angle BAD=angle CAE$, $angle PAE= angle NAD$.
$therefore triangle PAEsim triangle NAD$, whence, $frac{PA}{EA}=frac{AN}{AD}implies frac{1}{2}PA.ADsinangle{PAD}=frac{1}{2}EA.ANsinangle{EAN}$
Thus, $$Area(triangle PAD)=Area(triangle PAN)$$
$blacksquare$
answered Dec 7 '18 at 6:50
Anubhab GhosalAnubhab Ghosal
1,16919
answered Dec 7 '18 at 6:50
Anubhab GhosalAnubhab Ghosal
1,16919
answered Dec 7 '18 at 6:50
Anubhab GhosalAnubhab Ghosal
1,16919
answered Dec 7 '18 at 6:50
Anubhab GhosalAnubhab Ghosal
1,16919
1,16919
add a comment |
add a comment |
$begingroup$
Please include a diagram. [and some work you did]
$endgroup$
– coffeemath
Dec 6 '18 at 19:17
$begingroup$
@coffeemath Actually I tried to apply the formula for area using sin.I don't know nothing about P and N. That is my problem
$endgroup$
– rafa
Dec 6 '18 at 19:22
$begingroup$
Still, a diagram showing just A,B,C,D,E would be nice.
$endgroup$
– coffeemath
Dec 6 '18 at 19:23
$begingroup$
@coffeemath I don't know how to use it
$endgroup$
– rafa
Dec 6 '18 at 19:47