Prove that $BP,CQ,AD$ concur
$begingroup$
Triangle $ABC$ has an incircle $(I)$ which contacts $BC,CA,AB$ at $D,E,F$. On line $EF$ we get two points $M$ and $N$ such that $CM//BN//AD$. $DM$ and $DN$ cut $(I)$ at $P,Q$.
a, Prove that $BP,CQ,AD$ concur.
b, Let $J$ be point which $BP,CQ,AD$ concur. $X$ is midpoint of $PQ$. Show that $JX$ intersects $MN$ at the midpoint $G$ of $MN$.
I don't know which lemmas we use(maybe Ceva theorem, Thales theorem because there are three paralel lines). Show please and anyone can tell me some geometry book for studying? Thank.
geometry euclidean-geometry
edited Jan 1 at 6:08
Word Shallow
1,1182621
asked Jan 1 at 2:43
Quỳnh Vũ ThịQuỳnh Vũ Thị
274
$endgroup$
add a comment |
$begingroup$
Triangle $ABC$ has an incircle $(I)$ which contacts $BC,CA,AB$ at $D,E,F$. On line $EF$ we get two points $M$ and $N$ such that $CM//BN//AD$. $DM$ and $DN$ cut $(I)$ at $P,Q$.
a, Prove that $BP,CQ,AD$ concur.
b, Let $J$ be point which $BP,CQ,AD$ concur. $X$ is midpoint of $PQ$. Show that $JX$ intersects $MN$ at the midpoint $G$ of $MN$.
I don't know which lemmas we use(maybe Ceva theorem, Thales theorem because there are three paralel lines). Show please and anyone can tell me some geometry book for studying? Thank.
geometry euclidean-geometry
edited Jan 1 at 6:08
Word Shallow
1,1182621
asked Jan 1 at 2:43
Quỳnh Vũ ThịQuỳnh Vũ Thị
274
$endgroup$
$begingroup$
You mean $JX$ goes through $MN$ at $G$ and $G$ is midpoint of $MN$?
$endgroup$
– Word Shallow
Jan 1 at 4:27
$begingroup$
Yes, of course. help me show this pls.
$endgroup$
– Quỳnh Vũ Thị
Jan 1 at 5:42
add a comment |
$begingroup$
Triangle $ABC$ has an incircle $(I)$ which contacts $BC,CA,AB$ at $D,E,F$. On line $EF$ we get two points $M$ and $N$ such that $CM//BN//AD$. $DM$ and $DN$ cut $(I)$ at $P,Q$.
a, Prove that $BP,CQ,AD$ concur.
b, Let $J$ be point which $BP,CQ,AD$ concur. $X$ is midpoint of $PQ$. Show that $JX$ intersects $MN$ at the midpoint $G$ of $MN$.
I don't know which lemmas we use(maybe Ceva theorem, Thales theorem because there are three paralel lines). Show please and anyone can tell me some geometry book for studying? Thank.
geometry euclidean-geometry
edited Jan 1 at 6:08
Word Shallow
1,1182621
asked Jan 1 at 2:43
Quỳnh Vũ ThịQuỳnh Vũ Thị
274
$endgroup$
Triangle $ABC$ has an incircle $(I)$ which contacts $BC,CA,AB$ at $D,E,F$. On line $EF$ we get two points $M$ and $N$ such that $CM//BN//AD$. $DM$ and $DN$ cut $(I)$ at $P,Q$.
a, Prove that $BP,CQ,AD$ concur.
b, Let $J$ be point which $BP,CQ,AD$ concur. $X$ is midpoint of $PQ$. Show that $JX$ intersects $MN$ at the midpoint $G$ of $MN$.
I don't know which lemmas we use(maybe Ceva theorem, Thales theorem because there are three paralel lines). Show please and anyone can tell me some geometry book for studying? Thank.
geometry euclidean-geometry
geometry euclidean-geometry
edited Jan 1 at 6:08
Word Shallow
1,1182621
asked Jan 1 at 2:43
Quỳnh Vũ ThịQuỳnh Vũ Thị
274
edited Jan 1 at 6:08
Word Shallow
1,1182621
asked Jan 1 at 2:43
Quỳnh Vũ ThịQuỳnh Vũ Thị
274
edited Jan 1 at 6:08
Word Shallow
1,1182621
edited Jan 1 at 6:08
Word Shallow
1,1182621
edited Jan 1 at 6:08
Word Shallow
1,1182621
1,1182621
asked Jan 1 at 2:43
Quỳnh Vũ ThịQuỳnh Vũ Thị
274
asked Jan 1 at 2:43
Quỳnh Vũ ThịQuỳnh Vũ Thị
274
asked Jan 1 at 2:43
Quỳnh Vũ ThịQuỳnh Vũ Thị
274
274
$begingroup$
You mean $JX$ goes through $MN$ at $G$ and $G$ is midpoint of $MN$?
$endgroup$
– Word Shallow
Jan 1 at 4:27
$begingroup$
Yes, of course. help me show this pls.
$endgroup$
– Quỳnh Vũ Thị
Jan 1 at 5:42
add a comment |
$begingroup$
You mean $JX$ goes through $MN$ at $G$ and $G$ is midpoint of $MN$?
$endgroup$
– Word Shallow
Jan 1 at 4:27
$begingroup$
Yes, of course. help me show this pls.
$endgroup$
– Quỳnh Vũ Thị
Jan 1 at 5:42
$begingroup$
You mean $JX$ goes through $MN$ at $G$ and $G$ is midpoint of $MN$?
$endgroup$
– Word Shallow
Jan 1 at 4:27
$begingroup$
You mean $JX$ goes through $MN$ at $G$ and $G$ is midpoint of $MN$?
$endgroup$
– Word Shallow
Jan 1 at 4:27
$begingroup$
Yes, of course. help me show this pls.
$endgroup$
– Quỳnh Vũ Thị
Jan 1 at 5:42
$begingroup$
Yes, of course. help me show this pls.
$endgroup$
– Quỳnh Vũ Thị
Jan 1 at 5:42
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
a) Let $S$ be the intersection of $EF$ and $BC$,the segment $AD$ and incircle $(I)$ be $T$; $J$ be the intersection of $SP$ and $AD$; $BQ$ intersects $CP$ at $V$ .Then we have $ST$ is tangent of incircle $(I)$
So the polar of $S$ is the line $AD$.
We have: $V(SJ,QP)=-1$ and $V(SD,BC)=-1$
And $VSequiv VS,VBequiv VQ,VCequiv VPRightarrow VDequiv VJ$
So $Vin AD$. In $Delta VBC$: $P;Q$ are respectively in the $CV$ and $BV$
$PQ$ intersects $BC$ at $S$ and $(SD,BC)=-1$ so we have $MV;BQ;CQ$ concur.
Or $BP;CQ;AD$ concur $(Q.E.D)$
answered Jan 1 at 6:31
Word ShallowWord Shallow
1,1182621
$endgroup$
$begingroup$
What about exercise b? Thank you.
$endgroup$
– Quỳnh Vũ Thị
Jan 2 at 3:50
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3058186%2fprove-that-bp-cq-ad-concur%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
a) Let $S$ be the intersection of $EF$ and $BC$,the segment $AD$ and incircle $(I)$ be $T$; $J$ be the intersection of $SP$ and $AD$; $BQ$ intersects $CP$ at $V$ .Then we have $ST$ is tangent of incircle $(I)$
So the polar of $S$ is the line $AD$.
We have: $V(SJ,QP)=-1$ and $V(SD,BC)=-1$
And $VSequiv VS,VBequiv VQ,VCequiv VPRightarrow VDequiv VJ$
So $Vin AD$. In $Delta VBC$: $P;Q$ are respectively in the $CV$ and $BV$
$PQ$ intersects $BC$ at $S$ and $(SD,BC)=-1$ so we have $MV;BQ;CQ$ concur.
Or $BP;CQ;AD$ concur $(Q.E.D)$
answered Jan 1 at 6:31
Word ShallowWord Shallow
1,1182621
$endgroup$
$begingroup$
What about exercise b? Thank you.
$endgroup$
– Quỳnh Vũ Thị
Jan 2 at 3:50
add a comment |
$begingroup$
a) Let $S$ be the intersection of $EF$ and $BC$,the segment $AD$ and incircle $(I)$ be $T$; $J$ be the intersection of $SP$ and $AD$; $BQ$ intersects $CP$ at $V$ .Then we have $ST$ is tangent of incircle $(I)$
So the polar of $S$ is the line $AD$.
We have: $V(SJ,QP)=-1$ and $V(SD,BC)=-1$
And $VSequiv VS,VBequiv VQ,VCequiv VPRightarrow VDequiv VJ$
So $Vin AD$. In $Delta VBC$: $P;Q$ are respectively in the $CV$ and $BV$
$PQ$ intersects $BC$ at $S$ and $(SD,BC)=-1$ so we have $MV;BQ;CQ$ concur.
Or $BP;CQ;AD$ concur $(Q.E.D)$
answered Jan 1 at 6:31
Word ShallowWord Shallow
1,1182621
$endgroup$
$begingroup$
What about exercise b? Thank you.
$endgroup$
– Quỳnh Vũ Thị
Jan 2 at 3:50
add a comment |
$begingroup$
a) Let $S$ be the intersection of $EF$ and $BC$,the segment $AD$ and incircle $(I)$ be $T$; $J$ be the intersection of $SP$ and $AD$; $BQ$ intersects $CP$ at $V$ .Then we have $ST$ is tangent of incircle $(I)$
So the polar of $S$ is the line $AD$.
We have: $V(SJ,QP)=-1$ and $V(SD,BC)=-1$
And $VSequiv VS,VBequiv VQ,VCequiv VPRightarrow VDequiv VJ$
So $Vin AD$. In $Delta VBC$: $P;Q$ are respectively in the $CV$ and $BV$
$PQ$ intersects $BC$ at $S$ and $(SD,BC)=-1$ so we have $MV;BQ;CQ$ concur.
Or $BP;CQ;AD$ concur $(Q.E.D)$
answered Jan 1 at 6:31
Word ShallowWord Shallow
1,1182621
$endgroup$
a) Let $S$ be the intersection of $EF$ and $BC$,the segment $AD$ and incircle $(I)$ be $T$; $J$ be the intersection of $SP$ and $AD$; $BQ$ intersects $CP$ at $V$ .Then we have $ST$ is tangent of incircle $(I)$
So the polar of $S$ is the line $AD$.
We have: $V(SJ,QP)=-1$ and $V(SD,BC)=-1$
And $VSequiv VS,VBequiv VQ,VCequiv VPRightarrow VDequiv VJ$
So $Vin AD$. In $Delta VBC$: $P;Q$ are respectively in the $CV$ and $BV$
$PQ$ intersects $BC$ at $S$ and $(SD,BC)=-1$ so we have $MV;BQ;CQ$ concur.
Or $BP;CQ;AD$ concur $(Q.E.D)$
answered Jan 1 at 6:31
Word ShallowWord Shallow
1,1182621
answered Jan 1 at 6:31
Word ShallowWord Shallow
1,1182621
answered Jan 1 at 6:31
Word ShallowWord Shallow
1,1182621
answered Jan 1 at 6:31
Word ShallowWord Shallow
1,1182621
1,1182621
$begingroup$
What about exercise b? Thank you.
$endgroup$
– Quỳnh Vũ Thị
Jan 2 at 3:50
add a comment |
$begingroup$
What about exercise b? Thank you.
$endgroup$
– Quỳnh Vũ Thị
Jan 2 at 3:50
$begingroup$
What about exercise b? Thank you.
$endgroup$
– Quỳnh Vũ Thị
Jan 2 at 3:50
$begingroup$
What about exercise b? Thank you.
$endgroup$
– Quỳnh Vũ Thị
Jan 2 at 3:50
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3058186%2fprove-that-bp-cq-ad-concur%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
You mean $JX$ goes through $MN$ at $G$ and $G$ is midpoint of $MN$?
$endgroup$
– Word Shallow
Jan 1 at 4:27
$begingroup$
Yes, of course. help me show this pls.
$endgroup$
– Quỳnh Vũ Thị
Jan 1 at 5:42