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According to What is the temperature of heat generated from LPG gas?, the temperature of (1) LPG gas is above 1700 degree Celsius.

Many of us are familiar with pan-frying meat (steak, pan-frying the chicken breast etc).

Next, consider this: when you're (2) cooking meat at high temperatures ("above 200 °C) then it'll start forming heterocyclic amines (HCAs) and polycyclic aromatic hydrocarbons (PAHs), according to healthline.com and precisionnutrition.com.

Note: The way they're formed is when fat drips onto the flame or the utensil, which is at very high temperature, and then reacts with the high heat to form HCAs and PAHs.

Lastly, there is research that (3) corelates HCAs and PAHs with causing cancer in our bodies.

Sources: They're are all mentioned in the two articles that I linked above.

Question: If you combine these 3 facts, then we must be cooking a huge amount of HCAs + PAHs. It's a wonder that we aren't dead yet! How is it that we haven't developed cancer by eating meat cooked over a pan on an LPG gas (which I believe is quite common)? Or are we all consuming HCAs / PAHs slowly over time and will develop cancer? Or are one of the 3 facts above wrong?

Nothing with more thermal mass than a needle will reach the temperature of the flame. Note that if the stove or utensils would be this hot they would be glowing blinding white, while IRL they won't even glow a dim red. Granted if you dripped fat on a hot stove it would turn into a black tar containing PAH's, and a microscropic fraction of this could end up in your food. We are exposed to carcinogens and radiation all the time, theres no reason to be alarmed.

It is the temperature of the pan that matters not the flame

The flame temperature is irrelevant if you are cooking in a vessel. the only temperature you need to worry about is the temperature of the surface of the pan (or–even more importantly–the temperature of the meat). The surface of the pan will rarely get above around 220 °C if you are monitoring it. An unattended pan can get hotter, but, if you let it get too hot, your food will rapidly burn and will not be edible. Conversely, if the food is palatable, you probably haven't heated it enough to get lots of nasty HCAs and PAHs.

But the research that caused your worry is also frequently overstated. Yes, large quantities of PAHs or HCAs may be nasty, but the amounts in food–even food cooked on an open flame barbecue are very small and there have been no convincing studies showing a notable effect on health. There are some studies shooing a very small effect of meat on health (diseases such as bowel cancer have been linked to some meats but the studies are statistically weak and the effects are very small. Moreover these studies link to meat not cooking products but would probably have spotted any effects based on nasty cooking by-products). it is worth remembering that people have been cooking on open flames since we invented fire. If that were really dangerous, there would be strong evidence of harm and/or primitive man would have developed good defences to avoid the harm (as mild-drinkers evolved lactose tolerance in adults because their diets consisted of a lot of dairy products).

There have been many food scares based on observations of known nasties in cooked food. Acrolein and Nitites, for example. None of these have been shown to have any notable measurable effect on people in the concentrations present in the diet.