Determine-all-triangle-with-1-The-lengths-of-sides-positive-integers-and-at-least-one-is-prime-number-2-The-semiperimetr-is-positive-integer-and-area-is-equal-with-perimetr-

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Question Number 155100 by mathdanisur last updated on 25/Sep/21

$$\mathrm{Determine}\mathrm{all}\mathrm{triangle}\mathrm{with}:$$$$1.\mathrm{The}\mathrm{lengths}\mathrm{of}\mathrm{sides}\mathrm{positive}\mathrm{integers}$$$$\mathrm{and}\mathrm{at}\mathrm{least}\mathrm{one}\mathrm{is}\mathrm{prime}\mathrm{number}.$$$$2.\mathrm{The}\mathrm{semiperimetr}\mathrm{is}\mathrm{positive}\mathrm{integer}$$$$\mathrm{and}\mathrm{area}\mathrm{is}\mathrm{equal}\mathrm{with}\mathrm{perimetr}.$$

Commented by MJS_new last updated on 25/Sep/21

$$\mathrm{if}a<b<c\mathrm{there}\mathrm{are}\mathrm{only}4\mathrm{I}\mathrm{think}$$$$5/12/13$$$$6/25/29$$$$7/15/20$$$$9/10/17$$$$$$$$[{\mathrm{there}}^{\prime }\mathrm{s}\mathrm{a}{5}^{\mathrm{th}}\mathrm{one}\mathrm{without}\mathrm{primes}:6/8/10]$$

Commented by mathdanisur last updated on 25/Sep/21

$$\mathrm{perfect}\mathrm{my}\mathrm{dear}\mathrm{thankyou}$$

Answered by Rasheed.Sindhi last updated on 26/Sep/21

$$\bullet a,b,c,s=\frac{a+b+c}{2}\in {\mathbb{Z}}^{+}\wedge a\left(say\right)\in \mathbb{P}$$$$\bullet \underset{wheres=(a+b+c)/2}{\sqrt{s(s-a)(s-b)(s-c)}=a+b+c}$$$$\frac{a+b+c}{2}\in {\mathbb{Z}}^{+}\Rightarrow a+b+c\in \mathbb{E}$$$$\Rightarrow \{\begin{array}{l}a,b,callareeveeven\Rightarrow primeis2\\ \mathcal{T}woofa,b,careodd,thethirdiseven.\end{array}$$$$\mathrm{C}-1:a,b,c\in \mathbb{E}\Rightarrow a\left(prime\right)=2$$$$b=2m,c=2n;m,n\in {\mathbb{Z}}^{+}$$$$s=(2+2m+2n)/2=m+n+1$$$$s-a=1+m+n-2=m+n-1$$$$s-b=1+m+n-2m=-m+n+1$$$$s-c=1+m+n-2n=m-n+1$$$$\sqrt{s(s-a)(s-b)(s-c)}=a+b+c$$$$▸\sqrt{(m+n+1)(m+n-1)(-m+n+1)(m-n+1)}$$$$=2m+2n+2$$$$▸(m+n+1)(m+n-1)(-m+n+1)(m-n+1)$$$$=4{(m+n+1)}^2$$$$▸(m+n-1)(-m+n+1)(m-n+1)$$$$=4(m+n+1)$$$$(o,o)-case:m,n\in \mathbb{O}$$$$\overline{)\begin{array}{c}(o,o)-case:m,n\in \mathbb{O}\end{array}}$$$$\mathrm{RHS}isclearly\text{boldsymbol}even.$$$$\mathrm{LHS}:(o+o-o)(-o+o+o)(o-o+o)$$$$=(e-o)(e+o)(e+o)$$$$=\left(o\right)\left(o\right)\left(o\right)=o$$$$\mathrm{LHS}is\text{boldsymbol}odd.$$$$Contradiction.$$$$\overline{)\begin{array}{c}(e,e)-case:m,n\in \mathbb{E}\end{array}}$$$$\mathrm{RHS}=\text{boldsymbol}even$$$$\mathrm{LHS}:(e+e-o)(-e+e+o)(e-e+o)$$$$(e-o)(e+o)(e+o)$$$$\left(o\right)\left(o\right)\left(o\right)=o$$$$Contradiction.$$$$\overline{)\begin{array}{c}(e,o)or(o,e)-case:m\in \mathbb{E},n\in \mathbb{O}\end{array}}$$$$$$$$$$$$$$$$Continue$$

Commented by talminator2856791 last updated on 26/Sep/21

$$\mathrm{how}\mathrm{did}\mathrm{you}\mathrm{put}\mathrm{it}\mathrm{in}\mathrm{a}\mathrm{box}?$$

Commented by Rasheed.Sindhi last updated on 26/Sep/21

$$Themanuthatappearsonclick$$$$ofmatrix-buttoncontains\text{boldsymbol}table$$$$\text{boldsymbol}with\text{boldsymbol}borders.I^{\prime }vedeleteditsrow/s$$$$exceptonewhich{I}^{\prime }veusedasbox.$$

Commented by talminator2856791 last updated on 05/Oct/21

$$\mathrm{how}\mathrm{to}\mathrm{delete}\mathrm{the}\mathrm{rows}?$$

Commented by Rasheed.Sindhi last updated on 05/Oct/21

$$Whenyouareinarow(thatyouwant$$$$todelete)opensidemauandclick$$$${}^{\prime }delete{row}^{\prime }.$$

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